# Positively Curved Manifolds with Large Spherical Rank

Benjamin Schmidt  ,  Krishnan Shankar  and  Ralf Spatzier  Michigan State
University University of Oklahoma Univeristy of Michigan Dept. of Mathematics
Dept. of Mathematics Dept. of Mathematics 619 Red Cedar Road 601 Elm Avenue
530 Church Street East Lansing, MI, 48824 Norman, OK, 73019 Ann Arbor, MI,
48109<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Rigidity results are obtained for Riemannian $d$-manifolds with $\sec\geqslant
1$ and spherical rank at least $d-2>0$. Conjecturally, all such manifolds are
locally isometric to a round sphere or complex projective space with the
(symmetric) Fubini–Study metric. This conjecture is verified in all odd
dimensions, for metrics on $d$-spheres when $d\neq 6$, for Riemannian
manifolds satisfying the Rakić duality principle, and for Kählerian manifolds.

The first named author is partially supported by the NSF grant DMS–1207655.
The second named author is partially supported by the NSF grant DMS–1104352.
The third named author is partially supported by the NSF grant DMS–1307164.

##  1\. Introduction

A complete Riemannian $d$-manifold $M$ has extremal curvature
$\epsilon\in\\{-1,0,1\\}$ if its sectional curvatures satisfy
$\sec\leqslant\epsilon$ or $\sec\geqslant\epsilon$. For $M$ with extremal
curvature $\epsilon$, the rank of a complete geodesic
$\gamma:\mathbb{R}\rightarrow M$ is defined as the maximal number of linearly
independent, orthogonal, and parallel vector fields $V(t)$ along $\gamma(t)$
satisfying $\sec(\dot{\gamma},V)(t)\equiv\epsilon$. The manifold $M$ has
(hyperbolic, Euclidean or spherical according as $\epsilon$ is $-1,0$ or 1)
rank at least $k$ if all its complete geodesics have rank at least $k$.

Riemannian manifolds with $\sec\leqslant\epsilon$ and admitting positive rank
are known to be rigid. Finite volume Riemannian manifolds with bounded
nonpositive sectional curvatures and positive Euclidean rank are locally
reducible or locally isometric to symmetric spaces of nonpositive curvature
[1, 6]. Generalizations include [11] and [28]. Closed Riemannian manifolds
with $\sec\leqslant-1$ and positive hyperbolic rank are locally isometric to
negatively curved symmetric spaces [12]; this fails in infinite volume [8].
Finally, closed Riemannian manifolds with $\sec\leqslant 1$ and positive
spherical rank are locally isometric to positively curved, compact, rank one
symmetric spaces [25].

Rank rigidity results are less definitive in the $\sec\geqslant\epsilon$
curvature settings. Hyperbolic rank rigidity results for manifolds with
$-1\leqslant\sec\leqslant 0$ first appeared in [9]. Finite volume
$3$-manifolds with $\sec\geqslant-1$ and positive hyperbolic rank are real
hyperbolic [23]. Complete Riemannian $3$-manifolds with $\sec\geqslant 0$ and
positive Euclidean rank have reducible universal coverings as a special case
of [4], while the higher dimensional $\sec\geqslant 0$ examples in [26], [15]
illustrate that rank rigidity does not hold in complete generality.

Our present focus is the curvature setting $\sec\geqslant 1$. Conjecturally,
manifolds with $\sec\geqslant 1$ and positive spherical rank are locally
isometric to positively curved symmetric spaces. Note that the simply
connected, compact, rank one symmetric spaces, normalized to have minimum
sectional curvature 1, have spherical rank: $n-1=\dim(S^{n})-1$ for the
spheres; $2n-2=\dim(\mathbb{CP}^{n})-2$ for complex projective space;
$4n-4=\dim(\mathbb{HP}^{n})-4$ for quaternionic projective space;
$8=\dim(\mathbb{OP}^{2})-8$ for the Cayley projective plane. Our main theorems
concern $d$-manifolds with spherical rank at least $d-2$, spaces that are
conjecturally locally isometric to spheres or complex projective spaces.

######  Theorem A.

An odd dimensional Riemannian $d$-manifold with $d\geqslant 3$, $\sec\geqslant
1$, and spherical rank at least $d-2$ has constant sectional curvatures
$\sec\equiv 1$.

######  Theorem B.

Let $M$ be an even dimensional Riemannian $d$-manifold with $d\geqslant 4$,
$\sec\geqslant 1$, and spherical rank at least $d-2$. If $M$ does not have
constant sectional curvatures i.e., $\sec\not\equiv 1$, then $M$ satisfies:

  1. (1)

Every vector $v\in SM$ is contained in a $2$-plane section $\sigma$ with
$\sec(\sigma)>1$.

  2. (2)

The geodesic flow $\phi_{t}:SM\rightarrow SM$ is periodic with $2\pi$ a
period.

  3. (3)

There exists an almost complex structure $J:TM\rightarrow TM$ if $M$ is simply
connected.

  4. (4)

If $M$ is simply connected and if $\sec<9$, then every geodesic in $M$ is
simple, closed, and of length $\pi$. Moreover, $M$ is homotopy equivalent to
$\mathbb{C}\mathbb{P}^{d/2}.$

A Riemannian manifold satisfies the Rakić duality principle if for each $p\in
M$, orthonormal vectors $v,w\in S_{p}M$, and $c\in\mathbb{R}$, $v$ lies in the
$c$-eigenspace of the Jacobi operator $\mathcal{J}_{w}$ if and only if $w$
lies in the $c$-eigenspace of the Jacobi operator $\mathcal{J}_{v}$. This
property arises naturally in the study of Osserman manifolds [19, 20]. See
Section 2 for details.

######  Theorem C.

Let $M$ be a Riemannian $d$-manifold with $\sec\geqslant 1$ and spherical rank
at least $d-2$. If $M$ satisfies the Rakić duality principle, then $M$ is
locally symmetric.

######  Theorem D.

A Kählerian manifold with $\sec\geqslant 1$, real dimension $d\geqslant 4$,
and spherical rank at least $d-2$ is isometric to a symmetric
$\mathbb{C}\mathbb{P}^{d/2}$ with holomorphic curvatures equal to $4$.

Theorem A implies:

######  Corollary E.

A Riemannian $3$-manifold with $\sec\geqslant 1$ and positive spherical rank
has constant sectional curvatures.

Only the two- and six-dimensional spheres admit almost complex structures [5].
Hence, item (3) in Theorem B implies:

######  Corollary F.

A Riemannian sphere $S^{d}$ with $d\neq 2,6$, $\sec\geqslant 1$, and with
spherical rank at least $d-2$ has constant sectional curvatures.

It is instructive to compare the $\sec\geqslant 1$ case considered here with
that of the $\sec\leqslant 1$ case of rank-rigidity resolved in [25]. In both
cases, each unit-speed geodesic $\gamma:\mathbb{R}\rightarrow M$ admits a
Jacobi field $J(t)=\sin(t)V(t)$ where $V(t)$ is a normal parallel field along
$\gamma$ contributing to its rank . Hence, for each $p\in M$, the tangent
sphere of radius $\pi$ is contained in the singular set for
$\exp_{p}:T_{p}M\rightarrow M$. In a symmetric space with
$\frac{1}{4}\leqslant\sec\leqslant 1$, the first conjugate point along a unit-
speed geodesic occurs at time $\pi$, the soonest time allowed by the curvature
assumption $\sec\leqslant 1$. Consequently, the rank assumption is an
assumption about the locus of first singularities of exponential maps when
$\sec\leqslant 1$. In symmetric spaces with $1\leqslant\sec\leqslant 4$, the
first and second conjugate points along a unit-speed geodesic occur at times
$\pi/2$ and $\pi$, respectively. Therefore, when rank-rigidity holds in the
$\sec\geqslant 1$ setting, the rank assumption is an assumption about the
locus of second singularities of exponential maps. Concerning first
singularities, a simply-connected Riemannian manifold with $\sec\geqslant 1$
in which the first conjugate point along each unit-speed geodesic occurs at
time $\pi/2$ is globally symmetric [22].

An alternative definition for the spherical rank of a geodesic $\gamma$ in a
Riemannian manifold with $\sec\geqslant 1$ is the dimension of the space of
normal Jacobi fields along $\gamma$ that make curvature one with $\gamma$.
This alternative notion of rank is a priori less restrictive since parallel
fields $V(t)$ give rise to Jacobi fields $J(t)$ as described above. The Berger
spheres, suitably rescaled, have positive rank when defined in terms of Jacobi
fields [25] but not when defined in terms of parallel fields by Corollary E.
Moreover, there is an infinite dimensional family of Riemannian metrics on
$S^{3}$ with $\sec\geqslant 1$ and positive rank when defined in terms of
Jacobi fields [24]. In particular, there exists examples that are not locally
homogeneous. Each such metric admits a unit length Killing field $X$ with the
property that a $2$-plane section $\sigma\subset TM$ with $X\in\sigma$ has
$\sec(\sigma)=1$; the restriction of $X$ to a geodesic is a Jacobi field whose
normal component contributes to the rank. There are no known examples with
discrete isometry group.

To describe our methods and the organization of the paper, let
$\mathcal{I}=\\{p\in M\,|\,\sec_{p}\equiv 1\\}$ and
$\mathcal{O}=M\setminus\mathcal{I}$ denote the subsets of isotropic and
nonisotropic points in $M$, respectively. The goal is to prove that $M$ is
locally isometric to complex projective space when $\mathcal{O}\neq\emptyset.$

We start with a pointwise analysis of curvature one planes. Given a vector
$v\in S_{p}M$, let $E_{v}$ denote the span of all vectors $w$ orthogonal to
$v$ with $\sec(v,w)=1$ and let $D_{v}$ denote the subspace of $E_{v}$ spanned
by vectors contributing to the rank of the geodesic $\gamma_{v}(t)$. The
assignments $v\mapsto E_{v}$ and $v\mapsto D_{v}$ define two (possibly
singular) distributions on each unit tangent sphere $S_{p}M$, called the
eigenspace and spherical distributions, respectively (see 2.7 and 3.1). The
spherical rank assumption ensures that
$d-2=\dim(S_{p}M)-1\leqslant\dim(D_{v})$ for each $v\in S_{p}M$ so that both
distributions are of codimension at most one on $S_{p}M$.

The arrangement of curvature one planes at nonisotropic points $p$ encodes
what ought to be a complex structure, a source of rigidity. More precisely,
the eigenspace distribution on $S_{p}M$ is totally geodesic (see Lemma 2.12)
and of codimension at most one. Subsection 2.3 builds on earlier work of
Hangan and Lutz [13] where they exploited the fundamental theorem of
projective geometry to prove that codimension one totally geodesic
distributions on odd dimensional spheres are algebraic: there is a nonsingular
projective class $[A]$ of skew-symmetric linear maps of $\mathbb{R}^{n+1}$
with the property that the distribution is orthogonal to the Killing (line)
field on $S^{n}$ generated by $[A]$. In particular, such distributions are
projectively equivalent to the standard contact hyperplane distribution. Note
that when $M$ is complex projective space, with complex structure
$J:TM\rightarrow TM$, the codimension one eigenspace distribution on $S_{p}M$
is orthogonal to the Killing (line) field on $S_{p}M$ generated by $[J_{p}]$.

As the spherical distribution $D$ is invariant under parallel transport along
geodesics ($D_{\dot{\gamma}_{v}(t)}=P_{t}(D_{v})$), its study leads to more
global considerations in Section 3.1. The sphere of radius $\pi$ in $T_{p}M$
is also equipped with a kernel distribution, $v\mapsto
K_{v}:=\ker(\mathrm{d}(\exp_{p})_{v})$ (see 2.4). As each $w\in D_{v}$ is an
initial condition for an initially vanishing spherical Jacobi field along
$\gamma_{v}(t)$, parallel translation in $T_{p}M$ identifies the spherical
subspace $D_{v}$ with a subspace of $K_{\pi v}$ for each $v\in S_{p}M$ (see
Lemma 3.6). When $p\in\mathcal{O}$, the eigenvalue and spherical distributions
on $S_{p}M$ coincide (see Lemma 3.4). As a consequence, the kernel
distribution contains a totally geodesic subdistribution of codimension at
most one on $S(0,\pi)$. It follows that $\exp_{p}$ is constant on $S(0,\pi)$
(see Corollary 3.7) and that geodesics passing through nonisotropic points
$p\in\mathcal{O}$ are all closed (see Lemma 3.8). Moreover, when
$p\in\mathcal{O}$, each vector $v\in S_{p}M$ has rank exactly $d-2$ (see Lemma
3.12), or putting things together, the eigenspace distribution is a
nonsingular codimension one distribution on $S_{p}M$. As even dimensional
spheres do not admit such distributions, $M$ must have even dimension, proving
Theorem A. More generally, this circle of ideas and a connectivity argument
culminate in a proof that every vector in $M$ has rank $d-2$ when the
nonisotropic set $\mathcal{O}\neq\emptyset$ (see Proposition 3.13).

The remainder of the paper is largely based on curvature calculations in
radial coordinates with respected to frames adapted to the spherical
distributions that are introduced in Section 3.2. An argument based on these
calculations and the aforementioned fact that the spherical distributions are
contact distributions, establishes that if the nonisotropic set
$\mathcal{O}\neq\emptyset$ , then $M=\mathcal{O}$ (see Proposition 3.14). The
proof of Theorem B follows easily and appears in Section 3.3. The proof of
Theorem C appears in Section 3.4. There, the Rakić duality hypothesis is
applied to prove that the family of skew-symmetric endomorphisms
$A_{p}:T_{p}M\rightarrow T_{p}M$, $p\in M$, arising from the family of
eigenspace distributions on the unit tangent spheres $S_{p}M$, define an
almost complex structure on $M$ (see Lemma 3.22 ). This fact, combined with
additional curvature calculations in adapted framings, allows us to deduce
that $M$ is Einstein, from which the theorem easily follows (see the proof of
Proposition 3.21).

Finally, Sections 4 and 5 contain the proofs of Theorem D in real dimension at
least six and in real dimension four, respectively. The methods are largely
classical, relying on pointwise curvature calculations based on the Kähler
symmetries of the curvature tensor and on expressions for the curvature tensor
when evaluated on an orthonormal $4$-frame due to Berger [2, 17]. Essentially,
these calculations yield formulas that relate the eigenvalues of the
endomorphisms $A_{p}:T_{p}M\rightarrow T_{p}M$ to the curvatures of
eigenplanes in invariant four dimensional subspaces of $T_{p}M$. When the real
dimension is at least six, there are enough invariant four dimensional
subspaces to deduce that $M$ has constant holomorphic curvatures, concluding
the proof in that case. The argument in real dimension four proceeds
differently by proving that $M$ satisfies the Rakić duality principle. When
this fails, the decomposition of $TM$ into eigenplanes of $A:TM\rightarrow TM$
is shown to arise from a metric splitting of $M$, contradicting the curvature
hypothesis $\sec\geqslant 1$.

##  2\. Notation and Preliminaries

This section contains preliminary results, mostly well-known, that are used in
subsequent sections. Throughout $(M,\mathrm{g})$ denotes a smooth, connected,
and complete $d$-dimensional Riemannian manifold, $\mathcal{X}(M)$ the
$\mathbb{R}$-module of smooth vector fields on $M$, and $\nabla$ the Levi-
Civita connection. Let $X,Y,Z,W\in\mathcal{X}(M)$ be vector fields.
Christoffel symbols for the connection $\nabla$ are determined by Koszul’s
formula

$\displaystyle\mathrm{g}(\nabla_{X}Y,Z)$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\\{X\mathrm{g}(Y,Z)+Y\mathrm{g}(Z,X)-Z\mathrm{g}(X,Y)\\}$
$\displaystyle+$
$\displaystyle\tfrac{1}{2}\\{\mathrm{g}([X,Y],Z)-\mathrm{g}([Y,Z],X)+\mathrm{g}([Z,X],Y)\\}.$

The curvature tensor $R:\mathcal{X}(M)^{3}\rightarrow\mathcal{X}(M)$ is
defined by $R(X,Y)Z=[\nabla_{X},\nabla_{Y}]Z-\nabla_{[X,Y]}Z$ and has the
following symmetries

(2.2) $R(X,Y,Z,W)=-R(Y,X,Z,W)=R(Z,W,X,Y)$

where $R(X,Y,Z,W)=\mathrm{g}(R(X,Y)Z,W).$ The sectional curvature of a
$2$-plane section $\sigma$ spanned by vectors $v$ and $w$ is defined by
$\sec(\sigma)=\sec(v,w)=\frac{R(v,w,w,v)}{\|v\wedge w\|^{2}}$. An almost
Hermitian structure on $M$ is an almost complex structure $J:TM\rightarrow TM$
compatible with the metric: $\mathrm{g}(X,Y)=\mathrm{g}(JX,JY)$ for all
$X,Y\in\mathcal{X}(M)$. A Hermitian structure on $M$ consists of an integrable
almost Hermitian structure. The Kähler form is the $2$-form $\omega$ defined
by $\omega(X,Y)=\mathrm{g}(JX,Y)$. A Kähler structure on $M$ consists of a
Hermitian structure with closed Kähler form, $d\omega=0$, or equivalently, a
parallel complex structure, $\nabla J=0$. If $M$ is Kählerian, then
$\nabla_{Y}JX=J\nabla_{Y}X$ for all $X,Y\in\mathcal{X}(M)$, yielding the
additional curvature identities

$\displaystyle R(X,Y,Z,W)=R(JX,JY,Z,W)=$ (2.3) $\displaystyle
R(X,Y,JZ,JW)=R(JX,JY,JZ,JW).$

These curvature identities are the key properties of a Kählerian manifold used
in the proof of Theorem D.

###  2.1. Jacobi operators and eigenspace distributions.

Let $SM$ denote the unit sphere bundle of $M$; its fiber above a point $p\in
M$ is the unit sphere $S_{p}M$ in $T_{p}M$. For $v\in S_{p}M$ define the
Jacobi operator $\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}$ by
$\mathcal{J}_{v}(w)=R(w,v)v$. The symmetries (2.2) imply that
$\mathcal{J}_{v}$ is a well-defined self-adjoint linear map of $v^{\perp}$.
Its eigenvalues encode the sectional curvatures of $2$-plane sections
containing the vector $v$.

######  Lemma 2.1.

Let $v,w\in S_{p}M$ be orthonormal vectors and assume that
$\sec_{p}\geqslant\epsilon$ for some $\epsilon\in\mathbb{R}$. The following
are equivalent:

  1. (1)

$\sec(v,w)=\epsilon$

  2. (2)

$w$ is an eigenvector of $\mathcal{J}_{v}$ with eigenvalue $\epsilon$.

  3. (3)

$R(w,v)v=\epsilon w$

###### Proof.

Only $(1)\implies(2)$ is nontrivial. If $\\{e_{i}\\}_{i=1}^{n-1}$ is an
orthonormal eigenbasis of $\mathcal{J}_{v}$ with corresponding eigenvalues
$\lambda_{i}$, then $\lambda_{i}\geqslant\epsilon$ for each index $i$. Express
$w=\sum_{i=1}^{n-1}\alpha_{i}e_{i}$ with $\sum_{i=1}^{n-1}\alpha_{i}^{2}=1$.
Then
$\epsilon=\mathrm{g}(R(w,v)v,w)=\mathrm{g}(\mathcal{J}_{v}(w),w)=\sum_{i=1}^{n-1}\alpha_{i}^{2}\lambda_{i}.$
Conclude that $\alpha_{i}=0$ for indices $i$ with $\lambda_{i}>\epsilon$.
Therefore $w$ is an eigenvector of $\mathcal{J}_{v}$ with eigenvalue
$\epsilon$. ∎

######  Remark 2.2.

An analogous proof works when $\sec_{p}\leqslant\epsilon$.

######  Lemma 2.3.

Let $v,w\in S_{p}M$ be orthonormal vectors. If $w^{\perp}\cap v^{\perp}$
consists of eigenvectors of $\mathcal{J}_{v}$, then $w$ is an eigenvector of
$\mathcal{J}_{v}$. Consequently,
$R(v,w,w^{\prime},v)=\mathrm{g}(\mathcal{J}_{v}(w),w^{\prime})=0$ for any
$w^{\prime}\in w^{\perp}\cap v^{\perp}$.

###### Proof.

The orthogonal complement to an invariant subspace of a self-adjoint operator
is an invariant subspace. ∎

####  2.1.1. Specialization to manifolds with $\operatorname{cvc}(\epsilon)$.

######  Definition 2.4.

A Riemannian manifold has constant vector curvature $\epsilon$, denoted by
$\operatorname{cvc}(\epsilon)$, provided that $\epsilon$ is an extremal
sectional curvature for $M$ ($\sec\leqslant\epsilon$ or
$\sec\geqslant\epsilon$) and $\epsilon$ is an eigenvalue of $\mathcal{J}_{v}$
for each $v\in SM$ [23].

For each $v\in SM$, let $E_{v}\subset v^{\perp}$ denote the (nontrivial)
$\epsilon$-eigenspace of $\mathcal{J}_{v}$.

######  Convention 2.5.

For each $v\in S_{p}M$, parallel translation in $T_{p}M$ defines an
isomorphism between the subspace $v^{\perp}$ of $T_{p}M$ and the subspace
$T_{v}(S_{p}M)$ of $T_{v}(T_{p}M)$. This isomorphism is used without mention
when contextually unambiguous.

######  Convention 2.6.

Given a manifold $M$, an assignment $M\ni p\mapsto D_{p}\subset T_{p}M$ of
tangent subspaces is a distribution. The rank of the subspaces may vary with
$p\in M$ and the assignment is not assumed to have any regularity. The
codimension of a distribution $D$ is defined as the greatest codimension of
its subspaces. When a distribution $D$ is known to have constant rank, it is
called a nonsingular distribution.

######  Definition 2.7.

The $\epsilon$-eigenspace distribution on $S_{p}M$, denoted by $E$, is the
distribution of tangent subspaces

$S_{p}M\ni v\mapsto E_{v}\subset T_{v}(S_{p}M).$

Its regular set, denoted by $\mathcal{E}_{p}$, is the open subset of $S_{p}M$
consisting of unit vectors $v$ for which $\dim(E_{v})$ is minimal.

######  Example 2.8.

$\epsilon$-eigenspace distributions need not have constant rank. When $M$ is a
Berger sphere suitably rescaled to have $\operatorname{cvc}(1)$, the curvature
one $2$-planes in $S_{p}M$ are precisely the $2$-planes containing the Hopf
vector $h\in S_{p}M$. Therefore $\dim(E_{h})=\dim(E_{-h})=2$, while
$\dim(E_{v})=1$ for any vector $v\in S_{p}M\setminus\\{\pm h\\}$.

######  Lemma 2.9.

For each $p\in M$ the restriction of the $\epsilon$-eigenspace distribution on
$S_{p}M$ to $\mathcal{E}_{p}$ is smooth.

###### Proof.

The operators $\mathcal{J}_{v}-\epsilon\operatorname{Id}$ vary smoothly and
have constant rank in $\mathcal{E}_{p}$. Therefore the subspaces
$E_{v}=\ker(\mathcal{J}_{v}-\epsilon\operatorname{Id})$ vary smoothly with
$v\in\mathcal{E}_{p}$ (see [7, Lemma 1] for more details). ∎

######  Remark 2.10.

Let $\mathcal{E}\subset SM$ denote the collection of unit tangent vectors
$v\in SM$ with $\dim(E_{v})$ minimal. The same proof as that of Lemma 2.9
shows that the assignment $v\mapsto E_{v}$ is smooth on $\mathcal{E}$. Note
that $\mathcal{E}\cap S_{p}M$ may not coincide with $\mathcal{E}_{p}$.

A tangent distribution $D$ on a complete Riemannian manifold $S$ is totally
geodesic if complete geodesics of $S$ that are somewhere tangent to $D$ are
everywhere tangent to $D$.

######  Convention 2.11.

Henceforth, unit tangent spheres $S_{p}M$ are equipped with the standard
Riemannian metric, denoted by $\langle\cdot,\cdot\rangle$, induced from the
Euclidean metric $\mathrm{g}_{p}(\cdot,\cdot)$ on $T_{p}M$. Moreover,
geodesics in $S_{p}M$ are typically denoted by $c$ while geodesics in $M$ are
typically denoted by $\gamma$.

######  Lemma 2.12.

For each $p\in M$, the $\epsilon$-eigenspace distribution $E$ is a totally
geodesic distribution on $S_{p}M$.

###### Proof.

Let $v\in S_{p}M$ and $w\in E_{v}$. The geodesic $c(t)=\cos(t)v+\sin(t)w$
satisfies $c(0)=v$ and $\dot{c}(0)=w$. Calculate
$\mathcal{J}_{c(t)}(\dot{c}(t))=-\sin(t)\mathcal{J}_{w}(v)+\cos(t)\mathcal{J}_{v}(w)$.
By assumption, $\mathcal{J}_{v}(w)=\epsilon w$. By Lemma 2.1,
$\mathcal{J}_{w}(v)=\epsilon v$. Therefore
$\mathcal{J}_{c(t)}(\dot{c}(t))=\epsilon(-\sin(t)v+\cos(t)w)=\epsilon\dot{c}(t)$.
Hence $\dot{c}(t)\in E_{c(t)},$ concluding the proof. ∎

###  2.2. Conjugate points and Jacobi fields.

Let $M$ denote a smooth, connected, and complete Riemannian manifold.

######  Convention 2.13.

Henceforth, geodesics are parameterized by arclength. Moreover, the notation
$\gamma_{v}(t)$ is frequently used to denote a complete unit speed geodesic
with initial velocity $v=\dot{\gamma}_{v}(0)\in S_{\gamma(0)}M$.

Let $\exp_{p}:T_{p}M\rightarrow M$ denote the exponential map and
$\mathbf{r}:T_{p}M\setminus\\{0\\}\rightarrow S_{p}M$ the radial retraction
$\mathbf{r}(v)=\frac{v}{\|v\|}$. Critical points of $\exp_{p}$ are conjugate
vectors. For a conjugate vector $v\in T_{p}M$, let

(2.4) $K_{v}=\ker(\mathrm{d}(\exp_{p})_{v})\subset T_{v}(T_{p}M).$

The multiplicity of $v$ is defined as $\dim(K_{v})$. For $t>0$, let $S(0,t)$
denote the sphere in $T_{p}M$ with center $0$ and radius $t$. Gauss’ Lemma
asserts $K_{v}\subset T_{v}(S(0,\|v\|)).$

Let $v\in T_{p}M$ be a conjugate vector and
$\gamma(t)=\exp_{p}(t\mathbf{r}(v))$. The point $q=\exp_{p}(v)$ is conjugate
to the point $p$ along $\gamma$ at time $t=\|v\|$. The point $q=\exp_{p}(v)$
is a first conjugate point to $p$ along $\gamma$ if $v$ is a first conjugate
vector, i.e. $tv$ is not a conjugate vector for any $t\in(0,1)$. Denote the
locus of first conjugate vectors in $T_{p}M$ by $\operatorname{FConj}(p)$. The
conjugate radius at $p$, denoted $\operatorname{conj}(p)$, is defined by
$\operatorname{conj}(p)=\inf_{v\in\operatorname{FConj}(p)}\\{\|v\|\\}$ when
$\operatorname{FConj}(p)\neq\emptyset$ and $\operatorname{conj}(p)=\infty$
otherwise; when $\operatorname{FConj}(p)\neq\emptyset$, the infimum is
realized as a consequence of Lemma 2.14 below. The conjugate radius of $M$,
denoted $\operatorname{conj}(M)$, is defined by
$\operatorname{conj}(M)=\inf_{p\in M}\\{\operatorname{conj}(p)\\}$.

Equivalently, conjugate vectors and points are described in terms of Jacobi
fields along $\gamma$. A normal Jacobi field along $\gamma(t)$ is a vector
field $J(t)$, perpendicular to $\dot{\gamma}(t)$ and satisfying Jacobi’s
second order ode: $J^{\prime\prime}+R(J,\dot{\gamma})\dot{\gamma}=0.$ Initial
conditions $J(t),J^{\prime}(t)\in\dot{\gamma}(t)^{\perp}$ uniquely determine a
normal Jacobi field. Let $p=\gamma(0)$, $v=\dot{\gamma}(0)\in S_{p}M$, and
$w\in v^{\perp}$. The geodesic variation $\alpha(s,t)=\exp_{p}(t(v+sw))$ of
$\gamma(t)=\alpha(0,t)$ has variational field $J(t)=\frac{\partial}{\partial
s}\alpha(s,t)|_{s=0}$, a normal Jacobi field along $\gamma$ with initial
conditions $J(0)=0$ and $J^{\prime}(0)=w$ given by

(2.5) $J(t)=\mathrm{d}(\exp_{p})_{tv}(tw).$

If $J(a)=0$, then (2.5) implies that $aw\in K_{av}$. In this case $av$ is a
conjugate vector and $\gamma(a)$ is a conjugate point to $p=\gamma(0)$ along
$\gamma$. All initially vanishing normal Jacobi fields along $\gamma$ arise in
this fashion, furnishing the characterization: $\gamma(a)$ is conjugate to
$\gamma(0)$ along $\gamma$ if and only if there exists a nonzero normal Jacobi
field $J(t)$ along $\gamma$ with $J(0)=J(a)=0$.

For $\gamma(t)$ a geodesic and $t_{0}>0$, let $\mathcal{V}_{\gamma}^{t_{0}}$
denote the vector space of piecewise differentiable normal vector fields
$X(t)$ along $\gamma(t)$ with $X(0)=X(t_{0})=0$. The index form
$I_{\gamma}^{t_{0}}:\mathcal{V}_{\gamma}^{t_{0}}\times\mathcal{V}_{\gamma}^{t_{0}}\rightarrow\mathbb{R}$
is the bilinear symmetric map defined by

$I_{\gamma}^{t_{0}}(X,Y)=\int_{0}^{t_{0}}\mathrm{g}(X^{\prime},Y^{\prime})-\mathrm{g}(R(X,\dot{\gamma})\dot{\gamma},Y)\,dt.$

The null space of $I_{\gamma}^{t_{0}}$ consists of normal Jacobi fields $J(t)$
along $\gamma(t)$ with $J(0)=J(t_{0})=0$. By the Morse Index Theorem [10,
Chapter 11], there exists $X\in\mathcal{V}_{\gamma}^{t_{0}}$ such that
$I_{\gamma}^{t_{0}}(X,X)<0$ if and only if there exists $0<s<t_{0}$ such that
$\gamma(s)$ is conjugate to $\gamma(0)$ at time $s$. In particular, the
property of being a first conjugate point along a geodesic segment is a
symmetric property.

######  Lemma 2.14.

$\operatorname{FConj}(p)$ is a closed subset of $T_{p}M$.

###### Proof.

Assume that $v_{i}\in\operatorname{FConj}(p)$ converge to $v\in T_{p}M$. Let
$t_{i}=\|v_{i}\|$, $\bar{t}=\|v\|$, and
$w_{i}=\mathbf{r}(v_{i}),w=\mathbf{r}(v)\in S_{p}M$. As $v_{i}$ is a conjugate
vector, there exists a normal Jacobi field $J_{i}(t)$ along
$\gamma_{w_{i}}(t)$ with $J_{i}(0)=J_{i}(t_{i})=0$ and
$\|J_{i}^{\prime}(0)\|=1$. A subsequence of the Jacobi fields $J_{i}(t)$
converges to a nonzero Jacobi field $J(t)$ along $\gamma_{w}(t)$ with
$J(0)=J(\bar{t})=0$. Therefore $v$ is a conjugate vector. If
$v\notin\operatorname{FConj}(p)$, there exists $0<s<1$ such that $sv$ is a
conjugate vector. Therefore there exists
$X\in\mathcal{V}_{\gamma_{w}}^{\bar{t}}$ with
$I_{\gamma_{w}}^{\bar{t}}(X,X)<0$.

An orthonormal framing $\\{e_{1},\ldots,e_{n-1}\\}$ of a neighborhood $B$ of
$w$ in $S_{p}M$ induces parallel orthonormal framings
$\\{E_{1}(t),\ldots,E_{n-1}(t)\\}$ along geodesics with initial tangent
vectors in $B$, yielding isomorphisms between
$\mathcal{V}_{\gamma_{b}}^{t}\cong\mathcal{V}_{\gamma_{w}}^{t}$ for each $b\in
B$ and $t>0$. Under these isomorphisms,
$I_{\gamma_{w_{i}}}^{t_{i}}(X,X)\rightarrow I_{\gamma_{w}}^{\bar{t}}(X,X)$ by
continuity; therefore, $I_{\gamma_{w_{i}}}^{t_{i}}(X,X)<0$ for all $i$
sufficiently large, contradicting $v_{i}\in\operatorname{FConj}(p)$. ∎

###  2.3. Codimension one totally geodesic distributions on spheres.

Given a non-zero skew-symmetric linear map
$A:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ and $v\in S^{d-1}$, parallel
translation in $\mathbb{R}^{d}$ identifies $v^{\perp}$ and $T_{v}S^{d-1}$. As
$A$ is skew-symmetric and non-zero, the assignment $S^{d-1}\ni v\mapsto Av\in
T_{v}S^{d-1}$ defines a Killing field on $S^{d-1}$. Let
$E_{v}=\operatorname{span}\\{v,Av\\}^{\perp}$ denote the subspace of
$T_{v}S^{d-1}$ orthogonal to $Av$. Then $S^{d-1}\ni v\mapsto E_{v}\subset
T_{v}S^{d-1}$ defines a codimension one totally geodesic distribution on
$S^{d-1}$ with singular set $\mathcal{X}:=\\{x\in
S^{d-1}\,|\,E_{x}=T_{x}S^{d-1}\\}=\ker(A)\cap S^{d-1}$ as a consequence of the
following well-known lemma.

######  Lemma 2.15.

Let $X$ be a Killing field on a complete Riemannian manifold $(S,\mathrm{g})$.
If a geodesic $c(t)$ satisfies $\mathrm{g}(\dot{c},X)(0)=0$, then
$\mathrm{g}(\dot{c},X)(t)\equiv 0$.

The skew-symmetric linear map $A$ and each nonzero real multiple $rA$ yield
the same codimension one totally geodesic distribution $E$ on $S^{n}$. In
[13], Hangan and Lutz apply the fundamental theorem of projective geometry to
establish the following:

######  Theorem 2.16 (Hangan and Lutz).

Let $E$ be a nonsingular codimension one totally geodesic distribution on a
unit sphere $S^{d-1}\subset\mathbb{R}^{d}$. Then $d-1$ is odd and there exists
a nonsingular projective class $[A]\in PGL(\mathbb{R}^{d})$ of skew-symmetric
linear maps such that for each $x\in S^{d-1}$,
$E_{x}=\operatorname{span}\\{x,Ax\\}^{\perp}$.

The elegance of their approach lies in the fact that no a priori regularity
assumption is made, while a posteriori the distribution is algebraic. The
following corollary is immediate (see [13]).

######  Corollary 2.17.

A nonsingular codimension one totally geodesic distribution on an odd
dimensional unit sphere is real-analytic and contact.

######  Corollary 2.18.

Let $E$ be a nonsingular codimension one totally geodesic distribution on an
odd dimensional unit sphere $S^{d-1}$. The line field $L$ on $S^{d-1}$ defined
by $L=E^{\perp}$ is totally geodesic if and only if
$[A^{2}]=[-\operatorname{Id}]$, where $[A]$ is as in Theorem 2.16 above.

###### Proof.

Assume that $[A^{2}]=[-\operatorname{Id}]$. Choose a representative $A\in[A]$
with unit-modulus eigenvalues. Then $\|Av\|=1$ and $A^{2}v=-v$ for each $v\in
S^{d-1}$. The geodesic $c(t)=\cos(t)v+\sin(t)Av$ satisfies $\dot{c}(0)\in
L_{v}$. Then $\dot{c}(t)\in L_{c(t)}$ since
$\dot{c}(t)=-\sin(t)v+\cos(t)Av=Ac(t)$, concluding the proof that $L$ is
totally geodesic.

Conversely, assume that $L$ is totally geodesic. Let $v\in S^{d-1}$ and choose
a representative $A\in[A]$ satisfying $\|Av\|=1$. The geodesic
$c(t)=\cos(t)v+\sin(t)Av$ satisfies $\dot{c}(t)\in L_{c(t)}$ for each
$t\in\mathbb{R}$. Set $t=\frac{\pi}{2}$ and conclude that the $2$-plane
spanned by $v$ and $Av$ is invariant under $A$. As $\|Av\|=1$ and $A$ is skew-
symmetric, $A^{2}v=-v$, concluding the proof. ∎

Let $\mathcal{X}=\\{x\in S^{d-1}\,|\,E_{x}=T_{x}S^{d}\\}$ denote the singular
set for a codimension one totally geodesic distribution $E$ on $S^{d-1}$.
Given a subset $U\subset S^{d-1}$, let
$\Sigma(U)=\operatorname{span}\\{U\\}\cap S^{d-1}$ denote the smallest totally
geodesic subsphere of $S^{d-1}$ containing $U$.

######  Lemma 2.19.

The singular set $\mathcal{X}$ satisfies $\Sigma(\mathcal{X})=\mathcal{X}$.

###### Proof.

There is nothing to prove if $\mathcal{X}=\emptyset$. If $x\in\mathcal{X}$,
then $-x\in\mathcal{X}$ since each great circle through $-x$ also passes
through $x$. It remains to prove that for linearly independent
$x_{1},x_{2}\in\mathcal{X}$, the great circle
$C_{1}:=\operatorname{Sp}(\\{x_{1},x_{2}\\})\subset\mathcal{X}$.

If $x_{3}\in C_{1}\setminus\\{\pm x_{1},\pm x_{2}\\}$, then the line
$L_{1}:=T_{x_{3}}C_{1}$ is a subspace of $E_{x_{3}}$ since
$x_{1}\in\mathcal{X}$. Let $L_{2}$ be any other line in $T_{x_{3}}S^{d-1}$ and
let $C_{2}$ denote the great circle containing $x_{3}$ with tangent line
$L_{2}$.

Let $p\in C_{2}\setminus\\{\pm x_{3}\\}$. As $x_{1},x_{2}\in\mathcal{X}$ are
linearly independent, the tangent lines at $p$ to the great circles in the
totally geodesic $2$-sphere $\Sigma(C_{1}\cup C_{2})$ that join $x_{1}$ to $p$
and $x_{2}$ to $p$ are transverse subspaces of $E_{p}\cap
T_{p}\Sigma(C_{1}\cup C_{2})$. Therefore $T_{p}\Sigma(C_{1}\cup C_{2})\subset
E_{p}$. In particular, the tangent line to $C_{2}$ at $p$ is a subspace of
$E_{p}$, whence the line $L_{2}$ is a subspace of $E_{x_{3}}$, as required. ∎

######  Corollary 2.20.

The singular set $\mathcal{X}$ of a codimension one totally geodesic
distribution on $S^{d-1}$ does not contain a basis of $\mathbb{R}^{d}$.

The following simple lemma is applied to Riemannian exponential maps in
subsequent sections.

######  Lemma 2.21.

Let $E$ be a codimension one totally geodesic distribution on $S^{d-1}$, $X$ a
set, and $f:S^{d-1}\rightarrow X$ a function. If $f$ is constant on curves
everywhere tangent to $E$, then $f$ is constant.

###### Proof.

Let $x\in S^{d-1}$. The assumption implies that $f$ is constant on the union
of geodesics with initial velocity in $E_{x}$, a totally geodesic subsphere of
$S^{d-1}$ of codimension at most one. Any two such subspheres intersect. ∎

##  3\. Proofs of Theorems A, B, and C

Throughout this section $M$ denotes a complete $d$-dimensional Riemannian
manifold with $\sec\geqslant 1$ and spherical rank at least $d-2$. Then $M$ is
closed and has $\operatorname{cvc}(1)$. In particular, for each $v\in SM$, the
$1$-eigenspace $E_{v}$ of the Jacobi operator $\mathcal{J}_{v}$ (see
Definition 2.7) is a nonempty subspace of $v^{\perp}$.

Recall that a point $p\in M$ is isotropic if $\sec(\sigma)$ is independent of
the $2$-plane section $\sigma\subset T_{p}M$ and nonisotropic otherwise.
Hence, $p$ is an isotropic point if and only if $E_{v}=v^{\perp}$ for each
$v\in S_{p}M$. Let $\mathcal{I}$ and $\mathcal{O}$ denote the subsets of
isotropic and nonisotropic points in $M$, respectively. Note that
$\mathcal{I}$ is closed in $M$ and that $\mathcal{O}$ is open in $M$.

###  3.1. Preliminary structure and Proof of Theorem A.

This subsection discusses a number of preliminary structural results that
culminate in a proof of Theorem A.

For $p\in M$ and $v\in S_{p}M$, let $P_{t}:T_{p}M\rightarrow
T_{\gamma_{v}(t)}M$ denote parallel translation along the geodesic
$\gamma_{v}(t)$. Define the subspace $D_{v}\subset v^{\perp}$ by

$\displaystyle D_{v}$ $\displaystyle=$
$\displaystyle\operatorname{span}\\{w\in
v^{\perp}\,|\,\sec(\dot{\gamma_{v}}(t),P_{t}w)=1\,\,\,\,\,\forall
t\in\mathbb{R}\\}$ $\displaystyle=$ $\displaystyle\operatorname{span}\\{w\in
v^{\perp}\,|\,P_{t}w\in E_{\dot{\gamma}(t)}\,\,\,\,\,\forall
t\in\mathbb{R}\\}.$

Note that $D_{v}$ is a subspace of $E_{v}$ for each $v\in S_{p}M$. The
spherical rank assumption implies $\dim(D_{v})\geqslant d-2$. In particular,
the $1$-eigenspace distribution $E$ is a codimension one totally geodesic (by
Lemma 2.12) distribution on $S_{p}M$ when $p\in\mathcal{O}$.

######  Lemma 3.1.

For each $v\in S_{p}M$

  1. (1)

$D_{v}=D_{-v}$

  2. (2)

If $w\in D_{v}$, then $\sec(\dot{\gamma_{v}}(t),P_{t}w)=1$ for all
$t\in\mathbb{R}$.

###### Proof.

$(1)$ is immediate from the definition of $D_{v}$. For (2), let
$\\{u_{1},\ldots,u_{k}\\}$ be a maximal collection of linearly independent
vectors in

$\\{w\in v^{\perp}\,|\,P_{t}w\in E_{\dot{\gamma}(t)}\,\,\,\,\,\forall
t\in\mathbb{R}\\}$

and express $w=\sum_{i=1}^{k}a_{i}u_{i}$. As $E_{\dot{\gamma}(t)}$ is a
subspace, $P_{t}w=\sum_{i=1}^{k}a_{i}P_{t}u_{i}\in E_{\dot{\gamma}(t)}$ for
each $t\in\mathbb{R}$, concluding the proof. ∎

The rank of a vector $v\in S_{p}M$ is defined as $\dim(D_{v})$. The rank of a
one dimensional linear subspace $L\leqslant T_{p}M$ is defined as the rank of
a unit vector tangent to $L$. The rank of a geodesic is the common rank of
unit tangent vectors to the geodesic.

######  Definition 3.2.

The spherical distribution on $S_{p}M$, denoted by $D$, is the tangent
distribution defined by

$S_{p}M\ni v\mapsto D_{v}\subset T_{v}(S_{p}M).$

Let $\mathcal{D}_{p}$ denote the subset of $S_{p}M$ consisting of rank $d-2$
vectors and let $\mathcal{D}=\cup_{p\in M}\mathcal{D}_{p}$ denote the
collection of all rank $d-2$ unit vectors in $SM$.

As parallel translations along geodesics and sectional curvatures are
continuous, the rank of vectors cannot decrease under taking limits. This
implies the following:

######  Lemma 3.3.

For each $p\in M$, the regular set $\mathcal{D}_{p}$ is open and the spherical
distribution $D$ on $S_{p}M$ is continuous on its regular set
$\mathcal{D}_{p}$.

######  Lemma 3.4.

If $p$ is a nonisotropic point, i.e. $p\in\mathcal{O}$, then the spherical
distribution $D$ and eigenspace distribution $E$ coincide on $S_{p}M$.

###### Proof.

If not, then there exists a rank $d-2$ vector $v\in\mathcal{D}_{p}$ with the
property that $E_{v}=T_{v}(S_{p}M)$. Consider the codimension one totally
geodesic subsphere $S\subset S_{p}M$ containing $v$ and determined by
$T_{v}S=D_{v}$, namely $S=\exp_{v}(D_{v})$.

Given $x\in S_{p}M\setminus S$ sufficiently close to $v$, let $C(v,x)$ denote
the great circle through $v$ and $x$. Lemma 3.3 implies that the tangent line
$T_{x}C(v,x)$ is transverse to the subspace $D_{x}$. As $E$ is totally
geodesic and $E_{v}=T_{v}(S_{p}M)$, it follows that $T_{x}C(v,x)\subset
E_{x}$. Conclude that $E_{x}=T_{x}(S_{p}M)$. Lemma 2.20 implies that
$E=T(S_{p}M)$, a contradiction since $p\in\mathcal{O}$. ∎

######  Convention 3.5.

Parallel translation in $T_{p}M$ identifies the spherical distribution $D$ on
$S_{p}M$ with a distribution defined on the tangent sphere $S(0,\pi)\subset
T_{p}M$. The latter is also denoted by $D$ when unambiguous.

######  Lemma 3.6.

Let $v\in S_{p}M$. If $w\in D_{v}$, then $J(t)=\sin(t)P_{t}w$ is a Jacobi
field along $\gamma_{v}(t)$. In particular, $D_{\pi v}\subset K_{\pi v}$,
where $K_{\pi v}=\ker(\mathrm{d}(\exp_{p})_{\pi v})$ (see 2.4).

###### Proof.

Lemma 2.1 and Lemma 3.1(2) imply
$J^{\prime\prime}(t)+R(J,\dot{\gamma}_{v})\dot{\gamma}_{v}(t)=\sin(t)(-P_{t}w+R(P_{t}w,\dot{\gamma}_{v})\dot{\gamma}_{v}(t))=0$
for all $t\in\mathbb{R}$. As $J(\pi)=0$, $w\in K_{\pi v}$ by (2.5). ∎

######  Corollary 3.7.

If $p\in\mathcal{O}$, then the restriction of $\exp_{p}$ to the tangent sphere
$S(0,\pi)$ is a point map.

###### Proof.

The map $\exp_{p}$ is constant on curves tangent to the kernel distribution
defined by $S(0,\pi)\ni\pi v\mapsto K_{\pi v}\subset T_{\pi v}(S(0,\pi))$. The
distributions $E$ and $D$ coincide on $S_{p}M$ by Lemma 3.4. Lemma 3.6 implies
that $\exp_{p}$ is constant on curves tangent to the distribution $E$. Lemma
2.21 implies the corollary. ∎

Let $\phi_{t}:SM\rightarrow SM$, $t\in\mathbb{R}$, denote the geodesic flow.
For $T>0$, let

$\operatorname{Fix}_{T}=\\{v\in SM|\,\phi_{T}v=v\\}.$

######  Lemma 3.8.

If $p\in\mathcal{O}$, then $S_{p}M\subset\operatorname{Fix}_{2\pi}$.

###### Proof.

Corollary 3.7 implies $\exp_{p}(S(0,\pi))=\\{p^{\prime}\\}$ for some
$p^{\prime}\in M$. The lemma is a consequence of the Claim: for $v\in S_{p}M$,
the geodesics $\gamma_{v}(t)$ and $\gamma_{-v}(t)$ satisfy
$\dot{\gamma}_{v}(\pi)=-\dot{\gamma}_{-v}(\pi)$.

There exists a positive $\epsilon<\operatorname{inj}(M)$ such that
$\gamma_{v}(\epsilon)\in\mathcal{O}$ since $\mathcal{O}$ is open in $M$. Let
$w=\dot{\gamma}_{v}(\epsilon)$. Corollary 3.7 implies that
$q^{\prime}:=\gamma_{w}(\pi)=\gamma_{-w}(\pi)=\gamma_{-v}(\pi-\epsilon)$. The
geodesic segments $\gamma_{w}([\pi-\epsilon,\pi])$ and
$\gamma_{-v}([\pi-\epsilon,\pi])$ each have length $\epsilon$ and meet at the
points $p^{\prime}$ and $q^{\prime}$. As $\epsilon<\operatorname{inj}(M)$,
these segments coincide, implying the claim. ∎

######  Lemma 3.9.

Let $v\in S_{p}M$ have rank $d-2$ and let $w$ be a unit vector in
$v^{\perp}\cap D_{v}^{\perp}$. The initially vanishing normal Jacobi field
$J(t)$ along $\gamma_{v}(t)$ with $J(0)=0$ and $J^{\prime}(0)=w$ has the form
$J(t)=f(t)P_{t}w$ where $f(t)$ is the solution to the ODE
$f^{\prime\prime}+\sec(P_{t}w,\dot{\gamma}_{v})f=0$ with initial conditions
$f(0)=0$ and $f^{\prime}(0)=1$.

###### Proof.

The initial conditions $f(0)=0$ and $f^{\prime}(0)=1$ imply the initial
conditions $J(0)=0$ and $J^{\prime}(0)=w$. The hypotheses and Lemma 2.3 imply
that $P_{t}w$ is an eigenvector of $\mathcal{J}_{\dot{\gamma}_{v}(t)}$ with
eigenvalue $\sec(P_{t}w,\dot{\gamma}_{v})(t)$. Consequently,

$J^{\prime\prime}(t)+R(J,\dot{\gamma}_{v})\dot{\gamma_{v}}(t)=[f^{\prime\prime}(t)+\sec(P_{t}w,\dot{\gamma}_{v}(t))f(t)]P_{t}w=0$

concluding the proof. ∎

######  Corollary 3.10.

A vector $v\in S_{p}M$ has rank $d-1$ if and only if $\pi
v\in\operatorname{FConj}(p)$.

###### Proof.

If $v$ has rank $d-1$, then Lemma 3.6 implies that $\pi
v\in\operatorname{FConj}(p)$. If $v$ has rank $d-2$ then there is an initially
vanishing Jacobi field of the form described by Lemma 3.9. The function $f(t)$
vanishes strictly before $\pi$ by the equality case of the Rauch Comparison
Theorem [10, Chapter 11]. ∎

Recall that $\operatorname{FConj}(p)$ denotes the locus of first conjugate
vectors in $T_{p}M$.

######  Corollary 3.11.

If there exists $p\in M$ with $\operatorname{FConj}(p)=S(0,\pi)$, then
$M=\mathcal{I}$, i.e. $M$ has constant curvatures equal to one.

###### Proof.

Let $U_{p}=M\setminus\operatorname{Cut}(p)$. By Corollary 3.10, all vectors in
$S_{p}M$ have rank $d-1$. By Cartan’s theorem on determination of the metric
[10, Theorem 2.1, pg. 157], $U_{p}\subset\mathcal{I}$. Therefore,
$M=\operatorname{closure}({U}_{p})\subset\mathcal{I}.$ ∎

######  Lemma 3.12.

If $v\in S_{p}M$ has rank $d-1$ and the restriction of $\exp_{p}$ to
$S(0,\pi)$ is a point map, then $M=\mathcal{I}$.

###### Proof.

It suffices to prove $\operatorname{FConj}(p)=S(0,\pi)$ by Corollary 3.11. Let
$X=\operatorname{FConj}(p)\cap S(0,\pi)$. The vector $\pi v\in X$ by Corollary
3.10; therefore $X$ is a nonempty subset of $S(0,\pi)$. The subset $X$ is
closed in $S(0,\pi)$ by Lemma 2.14. It remains to demonstrate that $X$ is an
open subset of $S(0,\pi)$.

This fails only if there exists $x\in X$ and a sequence $x_{i}\in
S(0,\pi)\setminus X$ converging to $x$. As $\exp_{p}$ is a point map on
$S(0,\pi)$ each $x_{i}$ is a conjugate vector. As
$x_{i}\notin\operatorname{FConj}(p)$ there exists $s_{i}\in(0,1)$ such that
$s_{i}x_{i}\in\operatorname{FConj}(p)$. By Lemma 3.9, there exist Jacobi field
$J_{i}(t)=f_{i}(t)P_{t}w_{i}$ along the geodesics
$\gamma_{\mathbf{r}(x_{i})}(t)$ with $f_{i}(0)=f_{i}(s_{i})=f_{i}(\pi)=0$ for
each index $i$. Note that $\min\\{s_{i},\pi-s_{i}\\}>\operatorname{inj}(M)/2$.
Therefore, $s_{i}x_{i}$ converge to a conjugate vector $sx$ with $0<s<1$, a
contradiction. ∎

######  Proposition 3.13.

$SM=\mathcal{D}$ or $M=\mathcal{I}$.

###### Proof.

Assume that $\mathcal{I}$ is a proper subset of $M$, or equivalently, that
$\mathcal{O}\neq\emptyset$. Corollary 3.7 and Lemma 3.12 imply
$\mathcal{D}_{p}=S_{p}M$ for each $p\in\mathcal{O}$. Therefore
$\mathcal{D}\neq\emptyset$. As $\mathcal{D}$ is an open subset of the
connected $SM$, it remains to prove that $\mathcal{D}$ is a closed subset of
$SM$.

This fails only if there exists a sequence of rank $d-2$ vectors
$v_{i}\in\mathcal{D}$ with $v_{i}$ converging to a vector $v\in SM$ of rank
$d-1$. Lemma 3.8 implies each of the geodesics $\gamma_{v_{i}}$ is closed and
has $2\pi$ as a period; therefore, $\gamma_{v}$ is a closed geodesic having
$2\pi$ as a period. Let $p_{i}\in M$ denote the footpoint of each $v_{i}$ and
$p\in M$ the footpoint of $v$. As the rank of $v_{i}$ is $d-2$, the geodesic
$\gamma_{v_{i}}$ enters $\mathcal{O}$ at some time $t_{i}$. Replace $v_{i}$
with $w_{i}=\dot{\gamma}_{v_{i}}(t_{i})$. After possibly passing to a
subsequence, the sequence of rank $n-2$ vectors $w_{i}$ with footpoints
$q_{i}\in\mathcal{O}$ converge to a rank $d-1$ vector $w$ with footpoint $q$.

Continuity of $\exp:TM\rightarrow M$ and Lemma 3.7 imply that $\exp_{q}$
restricts to a point map on the tangent sphere $S(0,\pi)\subset T_{q}M$. Lemma
3.12 implies $M=\mathcal{I}$, a contradiction. ∎

Proof of Theorem A: Seeking a contradiction, assume that $M\neq\mathcal{I}$.
Then $SM=\mathcal{D}$ by Proposition 3.13. For $p\in M$, the spherical
distribution $D$ is a nonsingular codimension one tangent distribution on
$S_{p}M$, an even dimensional sphere since $M$ is odd dimensional. This
distribution is continuous by Lemma 3.3, a contradiction. $\Box$

###  3.2. Adapted Frames

This subsection consists of preliminary results that will culminate in the
proof of Theorem B in the next subsection. If $M$ does not have constant
curvatures equal to one, then Theorem A implies $d=\dim(M)$ is even and
Proposition 3.13 implies every tangent vector has rank $d-2$
($SM=\mathcal{D}$). These are standing assumptions on $M$ throughout this
subsection. The main result is the following proposition; its proof appears at
the end of this subsection.

######  Proposition 3.14.

If $M$ does not have constant curvatures equal to one, then $M$ has no
isotropic points ($M=\mathcal{O}$).

######  Lemma 3.15.

For each $p\in M$, the spherical distribution $D$ is a smooth tangent
distribution on $S_{p}M$.

###### Proof.

It suffices to prove smoothness of $D$ on a metric ball $B$ contained in the
tangent sphere $S_{p}M$. As the center $b_{0}$ of $B$ is a rank $d-2$ vector,
there exists a unit vector $w\in b_{0}^{\perp}$ and a $t_{0}>0$ such that
$\sec(\dot{\gamma}_{b_{0}}(t_{0}),P_{t_{0}}w)>1$. Therefore
$\gamma_{b_{0}}(t_{0})\in\mathcal{O}$, and since $\mathcal{O}$ is open,
$\gamma_{b}(t_{0})\in\mathcal{O}$ for all $b\in B$ after possibly reducing the
radius of $B$.

Lemma 3.4 implies $D_{\dot{\gamma}_{b}(t_{0})}=E_{\dot{\gamma}_{b}(t_{0})}$
for each $b\in B$. The unit tangent vectors $\dot{\gamma}_{b}(t_{0})$ vary
smoothly with $b\in B$. Remark 2.10 implies $D_{\dot{\gamma}_{b}(t_{0})}$
varies smoothly with $b\in B$. The lemma follows since $D_{b}$ is obtained by
parallel translating along $\gamma_{b}$ for time $t_{0}$ the subspace
$D_{\dot{\gamma}_{b}(t_{0})}$ to $T_{b}(S_{p}M)$. ∎

The proof of Proposition 3.14 is based on a curvature calculation in special
framings along geodesics. To introduce these framings, let $p\in M$, $v\in
S_{p}M$, and let $\\{e_{1},\ldots,e_{d-1}\\}\subset T_{v}(S_{p}M)$ be an
orthonormal basis with $e_{1},\ldots,e_{d-2}\in D_{v}$. Define
$E_{0}(t)=P_{t}v=\dot{\gamma}_{v}(t)$ for $t>0$ and $E_{i}(t)=P_{t}e_{i}$ for
$i\in\\{1,\ldots,d-1\\}$ and $t>0$.

######  Definition 3.16.

The parallel orthonormal framing $\\{E_{0}(t),\ldots,E_{d-1}(t)\\}$ along the
ray $\gamma_{v}:[0,\infty)\rightarrow M$ is an adapted framing.

The following describes curvature calculations in polar coordinates using
adapted framings.

Suppose that $B\subset S_{p}M$ is a metric ball of radius less than $\pi$.
Then $TB$ is trivial and the restriction of the spherical distribution $D$ to
$B$ is trivial. By Lemma 3.15, there are smooth unit length vector fields
$e_{1},\ldots,e_{d-2}$ on $B$ tangent to $D$. An orientation on $S_{p}M$
determines a positively oriented orthonormal framing
$\\{e_{1},\ldots,e_{d-1}\\}$ of $B$. For each $b\in B$, let
$\\{E_{0}(t),\ldots,E_{d-1}(t)\\}$ be the associated adapted framing along the
ray $\gamma_{b}$.

Now fix $v\in B$. For $T>0$ such that $Tv$ is not a conjugate vector,
$\exp_{p}$ carries a neighborhood $U$ of $Tv$ in $T_{p}M$ diffeomorphically
onto a neighborhood $V$ of $\exp_{p}(Tv)$ in $M$. After possibly reducing the
radius of $B$, the radial retraction of $U$ to the unit sphere $\textbf{r}(U)$
coincides with $B$. The collection of adapted framings along geodesic rays
with initial tangent vectors in $B$ restrict to an orthonormal framing
$\\{E_{0},\ldots,E_{d-1}\\}$ of the open set $V$ in $M$. To calculate the
Christoffel symbols in this framing, first define
$a_{ij}^{k}:B\rightarrow\mathbb{R}$ by

(3.1) $[e_{i},e_{j}]=\sum_{k=1}^{d-1}a_{ij}^{k}e_{k}.$

As $Tv$ is not a conjugate vector, the geodesic spheres $S(p,t)$ with center
$p$ and radius $t$ close to $T$ intersect the neighborhood $V$ in smooth
codimension one submanifolds. The vector fields $E_{1}(t),\ldots,E_{d-1}(t)$
are tangent to the distance sphere $S(p,t)$ in $V$ and have outward pointing
unit normal vector field $E_{0}(t)$. In what follows, $g^{\prime}:=E_{0}(g)$
denotes the radial derivative of a function $g$.

For each unit speed geodesic $\gamma(t)$ with initial velocity vector in $B$,
let $J_{i}(t)$ denote the Jacobi field along $\gamma$ with initial conditions
$J_{i}(0)=0$ and $J_{i}^{\prime}(0)=e_{i}\in T_{\dot{\gamma}(0)}(S_{p}M)$.
Lemmas 3.6 and 3.9 imply

$\displaystyle J_{i}(t)=\sin(t)E_{i}(t),$ $\displaystyle
i\in\\{1,\ldots,d-2\\},$ $\displaystyle J_{d-1}(t)=f(t)E_{d-1}(t),$

where $f(t)$ is the solution of the ODE

$f^{\prime\prime}+\sec(E_{0},E_{d-1})f=0,\quad\text{ with }\quad
f(0)=0,\,f^{\prime}(0)=1.$

For $t$ close to $T$, define $F_{t}:B\rightarrow M$ by
$F_{t}(b)=\exp_{p}(tb)$. The chain rule and (2.5) imply

(3.3) $\mathrm{d}F_{t}(e_{i})=J_{i}(t)$

for $i\in\\{1,2,\ldots,d-1\\}$. Use (3.1), (3.3), and the fact that the Jacobi
fields $J_{i}$ are invariant under the radial (geodesic) flow generated by
$E_{0}$ to deduce

(3.4) $\displaystyle[J_{i},J_{j}]=\sum_{k=1}^{d-1}a_{ij}^{k}J_{k},$
$\displaystyle\mathcal{L}_{E_{0}}J_{i}=[E_{0},J_{i}]=0.$

Use (3.2) and (3.4) to calculate that for $i,j\in\\{1,\ldots,d-2\\}$:

$\displaystyle[E_{0},E_{i}]=-\cot E_{i}$
$\displaystyle[E_{0},E_{d-1}]=\tfrac{-f^{\prime}}{f}E_{d-1}$ (3.5)
$\displaystyle[E_{i},E_{j}]=\sum_{k=1}^{d-2}\tfrac{a_{ij}^{k}}{\sin}E_{k}+\tfrac{a_{ij}^{d-1}f}{\sin^{2}}E_{d-1}$
$\displaystyle[E_{i},E_{d-1}]=\sum_{k=1}^{d-2}\tfrac{a_{i\,d-1}^{k}}{f}E_{k}+(\tfrac{a_{i\,d-1}^{d-1}}{\sin}-\tfrac{E_{i}(f)}{f})E_{d-1}.$

######  Lemma 3.17.

Let $i,j\in\\{1,\ldots,d-2\\}$. The orthonormal framing
$\\{E_{0},\ldots,E_{d-1}\\}$ has Christoffel symbols given by
$\nabla_{E_{0}}E_{k}=0$ for each $k\in\\{0,\ldots,d-2\\}$ and:

$\displaystyle\nabla_{E_{i}}E_{0}=\cot E_{i}$
$\displaystyle\nabla_{E_{i}}E_{j}=-\cot\delta_{i}^{j}E_{0}+\sum_{k=1}^{d-2}\tfrac{a_{ij}^{k}-a_{jk}^{i}+a_{ki}^{j}}{2\sin}E_{k}-\tfrac{1}{2}\\{\tfrac{a_{i\,d-1}^{j}+a_{j\,d-1}^{i}}{f}+\tfrac{a_{ji}^{d-1}f}{\sin^{2}}\\}E_{d-1}$
$\displaystyle\nabla_{E_{i}}E_{d-1}=\sum_{k=1}^{d-2}\tfrac{1}{2}\\{\tfrac{a_{i\,d-1}^{k}+a_{k\,d-1}^{i}}{f}+\tfrac{a_{ki}^{d-1}f}{\sin^{2}}\\}E_{k}$
$\displaystyle\nabla_{E_{d-1}}E_{0}=\tfrac{f^{\prime}}{f}E_{d-1}$
$\displaystyle\nabla_{E_{d-1}}E_{j}=\sum_{k=1}^{d-2}\tfrac{1}{2}\\{\tfrac{a_{k\,d-1}^{j}-a_{j\,d-1}^{k}}{f}-\tfrac{a_{jk}^{d-1}f}{\sin^{2}}\\}E_{k}+(\tfrac{E_{j}(f)}{f}-\tfrac{a_{j\,d-1}^{d-1}}{\sin})E_{d-1}$
$\displaystyle\nabla_{E_{d-1}}E_{d-1}=-\tfrac{f^{\prime}}{f}E_{0}+\sum_{k=1}^{d-2}(\tfrac{a_{k\,d-1}^{d-1}}{\sin}-\tfrac{E_{k}(f)}{f})E_{k}$

where $f(t)$ is the solution of the ODE

$f^{\prime\prime}+\sec(E_{0},E_{d-1})f=0,\quad\text{ with }\quad
f(0)=0,\,f^{\prime}(0)=1.$

###### Proof.

Calculate using (2) and (3.2). ∎

Use Lemma 3.17 to derive the curvature components: For
$i,j\in\\{1,\ldots,d-2\\}$,

(3.6)
$R(E_{i},E_{j},E_{0},E_{d-1})=\frac{-(a_{ij}^{d-1}f\csc)^{\prime}}{\sin}$
(3.7) $R(E_{d-1},E_{0},E_{j},E_{d-1})=\frac{(a_{j\,d-1}^{d-1}f\csc-
E_{j}(f))^{\prime}}{f}.$

Proof of Proposition 3.14: The goal is to prove $\mathcal{I}=M$ or
$\mathcal{I}=\emptyset$. The set of isotropic points $\mathcal{I}$ is closed
in $M$ and $M$ is connected. It suffices to prove that $\mathcal{I}$ is open
in $M$. Let $p\in\mathcal{I}$ and $v\in S_{p}M$. As $v$ has rank $d-2$, there
exists a positive $s<\pi$ such that $q:=\gamma_{v}(s)$ is the first conjugate
point to $p$ along the geodesic $\gamma_{v}(t)$.

Claim: $\mathcal{I}$ contains an open neighborhood of $q$ in $M$.

Assuming the claim holds, $\mathcal{I}$ contains an open neighborhood of the
point $p$ in $M$ since the property of being a first conjugate point along a
geodesic is symmetric. Hence $\mathcal{I}$ is open in $M$.

Proof of Claim: Let $w=-\dot{\gamma}_{v}(s)$ and note that $p=\gamma_{w}(s)$.
Let $B$ be a small metric ball in $S_{q}M$ containing $w$ and trivialize the
tangent bundle of $B$ with orthonormal vector fields
$\\{e_{1},\ldots,e_{d-1}\\}$ with $e_{i}(b)\in D_{b}$ for each $b\in B$ and
$i\in\\{1,\ldots,d-2\\}$. Consider the induced adapted framings
$\\{E_{0},\ldots,E_{d-1}\\}$ along geodesics with initial velocity vectors in
$B$.

If $q$ is not contained in an open neighborhood of isotropic points, then
there exists a sequence $q_{i}\in\mathcal{O}$ converging to $q$. As all
vectors have rank $d-2$ the spherical distributions on $S_{q_{i}}M$ converge
to the spherical distribution on $S_{q}M$.

As $q_{i}\in\mathcal{O}$, Lemma 3.4 implies that the spherical distribution on
each $S_{q_{i}}M$ is totally geodesic. Therefore, the limiting spherical
distribution on $S_{q}M$ is totally geodesic. By Corollary 2.17, the limiting
distribution on $S_{q}M$ is a contact distribution. In particular, the
function

$a_{12}^{d-1}=\langle[e_{1},e_{2}],e_{d-1}\rangle$

is nonzero on $B$. Use (3.6) to calculate

(3.8)
$R(E_{1},E_{2},E_{0},E_{d-1})(t)=\tfrac{a_{12}^{d-1}(w)}{\sin^{3}(t)}(\cos(t)f(t)-\sin(t)f^{\prime}(t))$

for $t\in(0,s)$ along $\gamma_{w}(t)$.

As $p\in\mathcal{I}$, the curvature tensor vanishes on orthonormal $4$-frames
at the point $p.$ Therefore as $t$ converges to $s$, the left hand side of
(3.8) converges to zero. As $a_{12}^{d-1}$ is nonzero on $B$, $(\cos f-\sin
f^{\prime})\rightarrow 0$ as $t\rightarrow s$.

Only the Jacobi field $J_{d-1}(t)$ can vanish before time $\pi$. As $p$ is
conjugate to $q$, $f(t)\rightarrow 0$ as $t\rightarrow s$. As $s<\pi$,
$\sin(s)\neq 0$. Conclude that $f(s)=f^{\prime}(s)=0$, a contradiction since
$J_{d-1}(t)=fE_{d-1}(t)$ is a nonzero Jacobi field along $\gamma_{w}(t)$.
$\Box$

###  3.3. Proof of Theorem B

Proof of (1): Let $v\in S_{p}M$. Since every tangent vector has rank $d-2$,
$\dim(D_{v})=\dim(v^{\perp})-1$. Proposition 3.14 and Lemma 3.4 imply
$D_{v}=E_{v}$. Lemma 2.1 concludes the proof. $\Box$

Proof of (2): Proposition 3.14 and Lemma 3.8 imply that
$SM\subset\operatorname{Fix}_{2\pi}$. $\Box$

Proof of (3): As in the proof of (1), $D_{v}=E_{v}$ and
$\dim(D_{v})=\dim(v^{\perp})-1=d-2$ for all $v\in SM$. Lemma 2.12 implies that
for each $p\in M$, the eigenspace distribution $E$ on $S_{p}M$ is a
nonsingular codimension one totally geodesic distribution. Theorem 2.16 yields
a nonsingular projective class $[A_{p}]\in PGL(T_{p}M)$ for each $p\in M$,
varying smoothly with $p\in M$ by Remark 2.10. For each $p\in M$ there are
precisely two representatives of the projective class $[A_{p}]$ having
determinant one. As $M$ is simply connected there exists a smooth section
$p\mapsto A_{p}\in[A_{p}]$. Item (3) is therefore a consequence of the polar
decomposition of $A_{p}$, see [3, Lemma 2.32, pg. 64] for details. $\Box$

The proof of item (4) of Theorem B requires some preliminary lemmas.

Corollary 3.7 and Proposition 3.14 imply that the restriction of $\exp_{p}$ to
the tangent sphere $S(0,\pi)\subset T_{p}M$ is a point map for each $p\in M$.
Define the map $F:M\rightarrow M$ by $F(p)=\exp_{p}(S(0,\pi))$. Then
$F^{2}=\operatorname{Id}$ by item (2) of Theorem B.

######  Lemma 3.18.

$F$ is an isometry of $M$.

###### Proof.

The map $F$ sends each complete geodesic in $M$ into itself while preserving
the lengths of subsegments. ∎

######  Lemma 3.19.

If $F$ has a fixed point, then $F=\operatorname{Id}$.

###### Proof.

By Lemma 3.18, it suffices to prove if $F(p)=p$, then the derivative map
$\mathrm{d}F_{p}=\operatorname{Id}$. The eigenvalues of the derivative map
$\mathrm{d}F_{p}$ are square roots of unity since $F^{2}=\operatorname{Id}$.
If $v\in T_{p}M$ is a unit length eigenvector of eigenvalue $-1$, then
$\mathrm{d}F_{p}(v)=\dot{\gamma}_{v}(\pi)=-v$. Therefore,
$\gamma_{v}(\pi+t)=\gamma_{v}(-t)$ for all $t$. By the chain rule,
$\dot{\gamma}_{v}(\pi+t)=-\dot{\gamma}_{v}(-t)$ for all $t$. When
$t=-\frac{\pi}{2}$ this implies
$\dot{\gamma}_{v}(\frac{\pi}{2})=-\dot{\gamma}_{v}(\frac{\pi}{2})$, a
contradiction. ∎

######  Lemma 3.20.

If $\sec<9$, then $F$ has a fixed point.

###### Proof.

If $F$ has no fixed points, then the displacement function of $F$, $x\mapsto
d(x,F(x))$, obtains a positive minimum value at some $p\in M$ as $M$ is
compact. A minimizing geodesic segment $\gamma$ that joints $p$ to $F(p)$ has
length $L\leqslant\operatorname{diam}(M)<\pi$ by Toponogov’s diameter rigidity
theorem [27] (see also [21, Remark 3.6, pg. 157]). Let $m$ denote the midpoint
of the segment $\gamma$. The union $\gamma\cup F(\gamma)$ forms a smoothly
closed geodesic of length $2L$ since otherwise $d(m,F(m))<L=d(p,F(p)$. By item
(2) and since $F$ has no fixed points, $2L\in\\{2\pi/(2k+1)\,|\,k\geqslant
1\\}$. Therefore, $\operatorname{inj}(M)\leqslant L\leqslant\pi/3$. As $M$ is
simply connected, even dimensional, and positively curved,
$\operatorname{inj}(M)=\operatorname{conj}(M)$. The Rauch comparison theorem
and the assumption $\sec<9$ imply that $\operatorname{conj}(M)>\pi/3$, a
contradiction. ∎

Proof of (4): Lemmas 3.19 and 3.20 imply that $F=\operatorname{Id}$. It
follows that each geodesic in $M$ is a closed geodesic having $\pi$ as a
period. If a closed geodesic of length $\pi$ is not simple, then there exist a
geodesic loop in $M$ of length at most $\pi/2$. In this case,
$\operatorname{inj}(M)\leqslant\pi/4$, contradicting
$\operatorname{inj}(M)=\operatorname{conj}(M)>\pi/3$. Therefore, each geodesic
in $M$ is simple, closed, and of length $\pi$.

Each unit speed geodesic starting at a point $p\in M$ of length $\pi$ has
equal index $k=1,3,7,$ or $\dim(M)-1$ in the pointed loop space $\Omega(p,p)$
by the Bott-Samelson Theorem [3, Theorem 7.23]. The multiplicity of each
conjugate point to $p$ in the interior of these geodesics is one since the
spherical Jacobi fields defined in Lemma 3.6 do not vanish before time $\pi$.
If $k\geqslant 3$, the Jacobi field given by Lemma 3.9 has a pair of
consecutive vanishing times $0<t_{1}<t_{2}<\pi$ satisfying
$t_{2}-t_{1}\leqslant\pi/k\leqslant\pi/3$. This contradicts
$\operatorname{conj}(M)>\pi/3$ as $\sec<9$. Conclude that $k=1$ and that $M$
has the homotopy type of $\mathbb{C}\mathbb{P}^{d/2}$ by [3, Theorem 7.23].
$\Box$

###  3.4. Proof of Theorem C

Recall that a Riemannian manifold satisfies the Rakić duality principle if for
each $p\in M$, orthonormal vectors $v,w\in S_{p}M$, and
$\lambda\in\mathbb{R}$, $v$ is a $\lambda$-eigenvector of the Jacobi operator
$\mathcal{J}_{w}$ if and only if $w$ is a $\lambda$-eigenvector of the Jacobi
operator $\mathcal{J}_{v}$. This subsection contains the proof of Theorem C,
an easy consequence of the next proposition.

######  Proposition 3.21.

Let $M$ be a complete and simply connected Riemannian $d$-manifold with
$d\geqslant 4$ even, $\sec\geqslant 1$, spherical rank at least $d-2$, and no
isotropic points. If $M$ satisfies the Rakić duality principle, then $M$ is
isometric to $\mathbb{C}\mathbb{P}^{d/2}$ endowed with the symmetric metric
having constant holomorphic curvatures equal to $4$.

The proof of this proposition appears at the end of the subsection. As a
preliminary step, observe that the proof of item (3) of Theorem B shows that
there exists a smooth section $p\mapsto A_{p}\in SL(T_{p}M)$ where each
$A_{p}$ is skew-symmetric and satisfies
$D_{v}=\operatorname{span}\\{v,A_{p}v\\}^{\perp}$ for each $v\in S_{p}M$.
Define $\lambda:SM\rightarrow\mathbb{R}$ by $\lambda(v)=\sec(v,A_{p}v)$ where
$p$ denotes the footpoint of the vector $v\in SM$.

######  Lemma 3.22.

$A_{p}^{2}=-\operatorname{Id}$ for each $p\in M$.

###### Proof.

The proof of item (1) of Theorem B shows that $A_{p}v$ is orthogonal to the
$1$-eigenspace $D_{v}$ of the Jacobi operator $\mathcal{J}_{v}$. Therefore
$\lambda(v)>1$ and $A_{p}v/\|A_{p}v\|$ is a unit vector in the
$\lambda(v)$-eigenspace of $\mathcal{J}_{v}$. Similarly,
$\lambda(A_{p}v/\|A_{p}v\|)>1$ and $A^{2}_{p}v/\|A_{p}^{2}v\|$ is a unit
vector in the $\lambda(A_{p}v/\|A_{p}v\|)$-eigenspace of the Jacobi operator
$\mathcal{J}_{A_{p}v/\|A_{p}v\|}$. The Rakić duality property implies that $v$
is a unit vector in the $\lambda(v)$-eigenspace of the Jacobi operator
$\mathcal{J}_{A_{p}v/\|A_{p}v\|}$. The Jacobi operator
$\mathcal{J}_{A_{p}v/\|A_{p}v\|}$ has two eigenspaces, the $1$-eigenspace
$D_{A_{p}v/\|A_{p}v\|}$ of dimension $d-2$ and its one dimensional orthogonal
complement, the $\lambda(A_{p}v/\|A_{p}v\|)$ eigenspace. Conclude that for
each $v\in S_{p}M$, $\lambda(v)=\lambda(A_{p}v/\|A_{p}v\|)$ and by skew-
symmetry of $A_{p}$ that $v=-A_{p}^{2}v/\|A^{2}_{p}v\|$. As $A_{p}^{2}v$ is a
multiple of $v$ for each $v\in S_{p}M$ and $A_{p}$ is skew-symmetric of
determinant one, $A_{p}^{2}=-\operatorname{Id}$. ∎

Fix $p\in M$ and a metric ball $B$ in the tangent sphere $S_{p}M$. Let
$\\{e_{1},\ldots,e_{d-1}\\}$ be a smooth framing of $B$ with
$\\{e_{1},\ldots,e_{d-2}\\}$ tangent to the spherical distribution $D$.

######  Lemma 3.23.

The field $e_{d-1}$ satisfies $\nabla_{e_{d-1}}e_{d-1}=0$ on $B$ with respect
to the round metric on $S_{p}M$. Equivalently, $a_{j\,d-1}^{d-1}=0$ for each
$j\in\\{1,\ldots,d-2\\}$.

###### Proof.

The first assertion is a consequence of Corollary 2.18 and Lemma 3.22. The
second is derived from (2)

$2\langle\nabla_{e_{d-1}}e_{d-1},e_{j}\rangle=\langle[e_{d-1},e_{d-1}],e_{j}\rangle-\langle[e_{d-1},e_{j}],e_{d-1}\rangle+\langle[e_{j},e_{d-1}],e_{d-1}\rangle=2a_{j\,d-1}^{d-1}.$

∎

Consider the adapted framing $\\{E_{0}(t),\ldots,E_{d-1}(t)\\}$ along
geodesics with initial velocity in $B$ induced by the framing
$\\{e_{1},\ldots,e_{d-1}\\}$ of $B$. Let $\epsilon<\operatorname{inj}(M)$ and
let $J(b,t)=f(b,t)E_{d-1}(t)$ be the Jacobi field along $\gamma_{b}(t)$
defined by Lemma 3.9. Then $f>0$ on $B\times(0,\epsilon)$.

######  Proposition 3.24.

The function $f:B\times(0,\epsilon)\rightarrow\mathbb{R}$ is radial:
$E_{j}(f)=0$ for each $j\in\\{1,\ldots,d-1\\}$, or equivalently, $f(b,t)$ does
not depend on $b\in B$.

###### Proof.

Lemma 3.23 and (3.7) imply
$R(E_{d-1},E_{0},E_{j},E_{d-1})=\frac{-E_{j}(f)^{\prime}}{f}$ for each
$j\in\\{1,\ldots,n-2\\}$. For each $b\in B$ and $t\in(0,\epsilon)$,
$E_{d-1}(b,t)$ is an eigenvector of eigenvalue $\lambda(E_{0}(b,t))$ for the
Jacobi operator $\mathcal{J}_{E_{0}(b,t)}$. The symmetry property implies that
$E_{0}(b,t)$ is an eigenvector of the Jacobi operator
$\mathcal{J}_{E_{d-1}(b,t)}$. Conclude $E_{j}(f)^{\prime}=0$ for each
$j\in\\{1,\ldots,d-2\\}$. Use (3.2) to calculate

$\displaystyle 0$ $\displaystyle=$ $\displaystyle E_{j}(f)^{\prime}$
$\displaystyle=$ $\displaystyle E_{0}E_{j}(f)$ $\displaystyle=$
$\displaystyle[E_{0},E_{j}](f)+E_{j}E_{0}(f)$ $\displaystyle=$
$\displaystyle-\cot E_{j}(f)+E_{j}(f^{\prime})$ $\displaystyle=$
$\displaystyle E_{j}(f^{\prime}-\cot f).$

Let $g=f^{\prime}-\cot f$. Corollary 2.17 and the fact that the time $t$-map
of the radial flow generated by $E_{0}$ carries the spherical distribution $D$
to the distribution spanned by $\\{E_{1}(t),\ldots,E_{d-2}(t)\\}$ on
$\exp_{p}(tB)\subset S(p,t)$ imply that the latter distribution is contact.
Conclude that $E_{d-1}(g)=0$ and that $g$ is a radial function.

Therefore

$h:=\frac{g}{\sin}=\frac{f^{\prime}\sin-\cos
f}{\sin^{2}}=(\frac{f}{\sin})^{\prime}$

is a radial function. Let $k=\frac{f}{\sin}$ and consider the restriction
$k(t)$ to a geodesic $\gamma_{b}(t)$ with $b\in B$. By L’Hopital’s rule and
the initial condition $f^{\prime}(0)=1$, $\lim_{t\rightarrow
0}k(t)=\frac{f^{\prime}(0)}{\cos(0)}=1$. By the fundamental theorem of
calculus, $k(t)=1+\int_{0}^{t}h(s)\,ds$ is a radial function. Therefore
$f=k\sin$ is a radial function. ∎

Proof of Proposition 3.21: It suffices to prove that
$\lambda:SM\rightarrow\mathbb{R}$ is constant by [7, Theorem 2, pg. 193]. Fix
$p\in M$ and a metric ball $B\subset S_{p}M$ as in Proposition 3.24.
Proposition 3.24 implies that $\lambda$ is constant on $B$ since by the Jacobi
equation, $\lambda(b)=\lim_{t\rightarrow 0}\frac{-f^{\prime\prime}}{f}(b,t)$
for each $b\in B$. As $S_{p}M$ is connected,
$\lambda:S_{p}M\rightarrow\mathbb{R}$ has a constant value $\lambda(p)$. Each
point $p\in M$ is an Einstein point with
$\operatorname{Ric}_{p}=(\lambda(p)+d-2)\mathrm{g}_{p}$. The adaptation of
Schur’s Theorem for Ricci curvatures [18, Note 3, Theorem 1, pg. 292] implies
that $M$ is globally Einstein. Therefore $\lambda(p)$ is independent of $p\in
M$. $\Box$

Proof of Theorem C: Apply Theorem A, Proposition 3.14, and Proposition 3.21.
$\Box$

##  4\. Proof of Theorem D in real dimension at least six

Throughout this section, $M$ is Kählerian with complex structure
$J:TM\rightarrow TM$, real even dimension $d\geqslant 4$, $\sec\geqslant 1$,
and spherical rank at least $d-2$. This section contains preliminary results,
culminating in the proof of Theorem D when $d\geqslant 6$.

As $M$ is orientable (complex), even-dimensional, and positively curved, $M$
is simply connected by Synge’s theorem. As $M$ is Kählerian, its second betti
number $b_{2}(M)\neq 0$, whence $M$ is not homeomorphic to a sphere. Therefore
$M$ does not have constant sectional curvatures.

Proposition 3.14 now implies that $M$ has no isotropic points
($M=\mathcal{O}$). Proposition 3.13 implies that every vector in $M$ has rank
$d-2$. Lemmas 2.12 and 3.4 imply that that the eigenspace distribution is a
nonsingular codimension one distribution on each unit tangent sphere in $M$.
By Theorem 2.16, there exists a nonsingular projective class $[A_{p}]\in
PGL(T_{p}M)$ of skew-symmetric maps such that
$D_{v}=E_{v}=\\{v,A_{p}v\\}^{\perp}$ for each $p\in M$ and $v\in S_{p}M$.

###  4.1. Relating the complex structure and the eigenspace distribution.

Fix $p\in M$ and choose a representative $A_{p}\in[A_{p}]$. Assume that
$V=\sigma_{1}\oplus\sigma_{2}$ is an orthogonal direct sum of two
$A_{p}$-invariant $2$-plane sections. There exist scalars $0<\mu_{1}$ and
$0<\mu_{2}$ such that $\|A_{p}v_{i}\|=\mu_{i}$ for each unit vector
$v_{i}\in\sigma_{i}$. There is no loss in generality in assuming
$\mu_{1}\leqslant\mu_{2}$ and if equality $\mu_{1}=\mu_{2}$ holds, then
$\lambda_{1}\leqslant\lambda_{2}$.

For a unit vector $v\in S_{p}M$, let $\lambda(v)=\sec(v,A_{p}v)$. Then
$A_{p}v$ is an eigenvector of the Jacobi operator $\mathcal{J}_{v}$ with
eigenvalue $\lambda(v)>1$. Note that $\lambda(v)$ is the maximal curvature of
a $2$-plane section containing the vector $v$. Therefore,
$\lambda(A_{p}v/\|A_{p}v\|)\geqslant\lambda(v)$ with equality only if
$A_{p}^{2}v$ and $v$ are linearly dependent. For a vector
$v_{i}\in\sigma_{i}$, let $\bar{v}_{i}=A(v_{i})/\mu_{i}$. With this notation,
$\bar{\bar{v_{i}}}=-v_{i}$.

######  Lemma 4.1.

Assume that $\\{u,v,w\\}\subset V$ are orthonormal vectors with
$u,v\in\sigma_{i}$ and $w\in\sigma_{j}$ with $i\neq j\in\\{1,2\\}$. Then
$R(u,v,w,u)=0$ and $R(u,w,w,u)=1$.

###### Proof.

As $u\in\sigma_{i}$, an $A_{p}$-invariant $2$-plane, the orthogonal $2$-plane
$\sigma_{j}$ is contained in $E_{u}$. In particular, $w\in E_{u}$, implying
the lemma. ∎

######  Lemma 4.2.

Let $v_{i}\in\sigma_{i}$, $i=1,2$, be unit-vectors. If $v=av_{1}+bv_{2}$ is a
unit-vector, then $\lambda(v)=a^{2}\lambda_{1}+b^{2}\lambda_{2}$.

###### Proof.

Observe that $\lambda(v)\|A_{p}v\|^{2}=R(v,A_{p}v,A_{p}v,v)$ or equivalently

$\lambda(v)(a^{2}\mu_{1}^{2}+b^{2}\mu_{2}^{2})=R(av_{1}+bv_{2},a\mu_{1}\bar{v}_{1}+b\mu_{2}\bar{v}_{2},a\mu_{1}\bar{v}_{1}+b\mu_{2}\bar{v}_{2},av_{1}+bv_{2}).$

Expanding the above, using Lemma 4.1, and simplifying yields

(4.1)
$\lambda(v)(a^{2}\mu_{1}^{2}+b^{2}\mu_{2}^{2})=a^{4}\mu_{1}^{2}\lambda_{1}+a^{2}b^{2}(\mu_{1}^{2}+\mu_{2}^{2})+b^{4}\mu_{2}^{2}\lambda_{2}+\Phi$

where

(4.2)
$\Phi=2a^{2}b^{2}\mu_{1}\mu_{2}[R(v_{1},\bar{v}_{1},\bar{v}_{2},v_{2})+R(v_{1},\bar{v}_{2},\bar{v}_{1},v_{2})].$

The vector $w:=b\mu_{2}\bar{v}_{1}-a\mu_{1}\bar{v}_{2}$ is orthogonal to both
$v$ and $A_{p}v$ so that $1=\sec(v,w)$. Equivalently

$(a^{2}\mu_{1}^{2}+b^{2}\mu_{2}^{2})=R(av_{1}+bv_{2},b\mu_{2}\bar{v}_{1}-a\mu_{1}\bar{v}_{2},b\mu_{2}\bar{v}_{1}-a\mu_{1}\bar{v}_{2},av_{1}+bv_{2}).$

Expanding the above, using Lemma 4.1, and simplifying yields

(4.3)
$\Phi=a^{2}b^{2}(\mu_{2}^{2}\lambda_{1}+\mu_{1}^{2}\lambda_{2})+a^{2}\mu_{1}^{2}(a^{2}-1)+b^{2}\mu_{2}^{2}(b^{2}-1).$

Substituting (4.3) into (4.1) and simplifying using $a^{2}+b^{2}=1$ yields the
desired formula for $\lambda(v)$. ∎

######  Corollary 4.3.

If $\mu_{1}<\mu_{2}$, then $\lambda_{1}<\lambda_{2}$.

###### Proof.

In the notation of Lemma 4.2, choose the vector $v$ so that $a=b=\sqrt{2}/2$.
As $\mu_{1}<\mu_{2}$, the vectors $v=av_{1}+bv_{2}$ and
$A_{p}^{2}v=-(a\mu_{1}^{2}v_{1}+b\mu_{2}^{2}v_{2})$ are linearly independent.
Therefore
$\lambda(v)=\sec(v,A_{p}v)<\sec(A_{p}v,A_{p}^{2}v)=\lambda(A_{p}v/\|A_{p}v\|)$.
By Lemma 4.2,
$\frac{1}{2}\lambda_{1}+\frac{1}{2}\lambda_{2}<\frac{\mu_{1}^{2}}{\mu_{1}^{2}+\mu_{2}^{2}}\lambda_{1}+\frac{\mu_{2}^{2}}{\mu_{1}^{2}+\mu_{2}^{2}}\lambda_{2}$,
or equivalently,
$(\frac{1}{2}-\frac{\mu_{1}^{2}}{\mu_{1}^{2}+\mu_{2}^{2}})\lambda_{1}<(\frac{\mu_{2}^{2}}{\mu_{1}^{2}+\mu_{2}^{2}}-\frac{1}{2})\lambda_{2}$.
If $\lambda_{2}\leqslant\lambda_{1}$, it follows that
$\frac{1}{2}-\frac{\mu_{1}^{2}}{\mu_{1}^{2}+\mu_{2}^{2}}<\frac{\mu_{2}^{2}}{\mu_{1}^{2}+\mu_{2}^{2}}-\frac{1}{2}$,
a contradiction. ∎

Given unit vectors $e_{i}\in\sigma_{i}$, $i=1,2$, consider the following
components of the curvature tensor:
$\alpha=R(e_{1},\bar{e}_{1},e_{2},\bar{e}_{2})$,
$b=R(\bar{e}_{1},e_{2},e_{1},\bar{e}_{2})$, and
$\gamma=R(e_{2},e_{1},\bar{e}_{1},\bar{e}_{2})$. By the Bianchi identity,

(4.4) $\alpha+\beta+\gamma=0.$

######  Lemma 4.4.

In the notation above, $\beta=\gamma>0$, $\alpha=-2\gamma<0$, and
$\mu_{1}^{2}(\lambda_{2}-1)+\mu_{2}^{2}(\lambda_{1}-1)=6\mu_{1}\mu_{2}\gamma$.
Moreover, $(\lambda_{1}-1)(\lambda_{2}-1)\leqslant 9\gamma^{2}$.

###### Proof.

Set $v_{1}=e_{1}$, $v_{2}=e_{2}$, and $a=b=\frac{\sqrt{2}}{2}$ and use (4.2)
and (4.3) to deduce

(4.5)
$\mu_{1}^{2}(\lambda_{2}-1)+\mu_{2}^{2}(\lambda_{1}-1)=2\mu_{1}\mu_{2}(\beta-\alpha).$

after simplification.

Similarly, set $v_{1}=e_{1}$, $v_{2}=\bar{e}_{2}$, and
$a=b=\frac{\sqrt{2}}{2}$ and use (4.2) and (4.3) to deduce

(4.6)
$\mu_{1}^{2}(\lambda_{2}-1)+\mu_{2}^{2}(\lambda_{1}-1)=2\mu_{1}\mu_{2}(\gamma-\alpha).$

As $\mu_{i}>0$ and $\lambda_{i}>1$, (5.3) and (4.6) imply that $\beta=\gamma$.
By (4.4) $\alpha=-2\gamma$ which upon substitution into (4.6) yields

$\mu_{1}^{2}(\lambda_{2}-1)+\mu_{2}^{2}(\lambda_{1}-1)=6\mu_{1}\mu_{2}\gamma$

from which the remaining inequalities are easily deduced. ∎

######  Lemma 4.5.

In the notation above, $\lambda_{1}\leqslant 3\gamma+1\leqslant\lambda_{2}$.
Equality holds in either case only if $\lambda_{1}=\lambda_{2}=3\gamma+1$ and
$\mu_{1}=\mu_{2}.$

###### Proof.

If $\lambda_{1}\geqslant 3\gamma+1$, then
$9\gamma^{2}\leqslant(\lambda_{1}-1)^{2}\leqslant(\lambda_{1}-1)(\lambda_{2}-1)\leqslant
9\gamma^{2}$, implying that $\lambda_{1}=\lambda_{2}=3\gamma+1$ (and
$\mu_{1}=\mu_{2}$ by Corollary 4.3). Lemma 4.4 and the derivation of Berger’s
curvature inequality [2, 17] imply

$\displaystyle 2\gamma=-\alpha=R(\bar{e}_{1},e_{1},e_{2},\bar{e}_{2})=$
$\displaystyle\frac{1}{6}[\sec(\bar{e}_{1}+\bar{e}_{2},e_{1}+e_{2})+\sec(e_{1}+\bar{e}_{2},\bar{e}_{1}-e_{2})]$
$\displaystyle+\frac{1}{6}[\sec(\bar{e}_{1}-\bar{e}_{2},e_{1}-e_{2})+\sec(e_{1}-\bar{e}_{2},\bar{e}_{1}+e_{2})]$
$\displaystyle-\frac{1}{6}[\sec(\bar{e}_{1}-\bar{e}_{2},e_{1}+e_{2})+\sec(e_{1}-\bar{e}_{2},\bar{e}_{1}-e_{2})]$
$\displaystyle-\frac{1}{6}[\sec(\bar{e}_{1}+\bar{e}_{2},e_{1}-e_{2})+\sec(e_{1}+\bar{e}_{2},\bar{e}_{1}+e_{2})].$

If $\sigma\subset V=\sigma_{1}\oplus\sigma_{2}$ is a $2$-plane section and
$v\in\sigma$ is a unit vector, then
$\sec(\sigma)\leqslant\lambda(v)\leqslant\lambda_{2}$ where the last
inequality is a consequence of Lemma 4.2. Hence
$1\leqslant\sec\leqslant\lambda_{2}$ on $V$. These inequalities and the above
formula for $2\gamma$ yields the inequality $\lambda_{2}\geqslant 3\gamma+1$
where equality holds only if
$\sec(\bar{e}_{1}+\bar{e}_{2},e_{1}+e_{2})=\lambda_{2}$. Hence, equality holds
only if
$\lambda_{2}=\sec(\bar{e}_{1}+\bar{e}_{2},e_{1}+e_{2})\leqslant\lambda((e_{1}+e_{2})/\sqrt{2})=\frac{1}{2}(\lambda_{1}+\lambda_{2})\leqslant\lambda_{2}$,
or equivalently if $\lambda_{1}=\lambda_{2}$ (and $\mu_{1}=\mu_{2}$ by
Corollary 4.3). ∎

######  Lemma 4.6.

For each nonzero vector $v\in T_{p}M$, there exists
$c(v)\in\mathbb{R}\setminus\\{0\\}$ such that $A_{p}J_{p}v=c(v)J_{p}A_{p}v$.

###### Proof.

Let $v\in T_{p}M\setminus\\{0\\}$. Lemma 2.1 and (2.3) imply that
$J_{p}(E_{v})=E_{J_{p}v}$. Therefore

$\displaystyle\operatorname{span}\\{J_{p}v,E_{J_{p}v}\\}=\operatorname{span}\\{J_{p}v,J_{p}(E_{v})\\}=$
$\displaystyle
J_{p}(\operatorname{span}\\{v,E_{v}\\})=J_{p}((A_{p}v)^{\perp})=J_{p}(A_{p}v)^{\perp},$

where the last equality uses the fact that $J_{p}$ acts orthogonally. Conclude
that both the vectors $J_{p}A_{p}v$ and $A_{p}J_{p}v$ are perpendicular to the
codimension one subspace $\operatorname{span}\\{J_{p}v,E_{J_{p}v}\\}$,
concluding the proof. ∎

######  Lemma 4.7.

Either $A_{p}J_{p}=J_{p}A_{p}$ or $A_{p}J_{p}=-J_{p}A_{p}$.

###### Proof.

As both $A_{p}J_{p}$ and $J_{p}A_{p}$ are non-degenerate, Lemma 4.6 implies
that there is a nonzero constant $c\in\mathbb{R}$ such that
$A_{p}J_{p}=cJ_{p}A_{p}$. Taking the determinant yields $c^{d}=1$, whence
$c=\pm 1$ since $d$ is even. ∎

######  Proposition 4.8.

$A_{p}J_{p}=J_{p}A_{p}$.

###### Proof.

Let $\sigma_{1}$ be an $A_{p}$-invariant $2$-plane section. If $\sigma_{1}$ is
$J_{p}$-invariant, then the restriction of $A_{p}$ and $J_{p}$ to $\sigma_{1}$
differ by a scalar, hence commute, concluding the proof in this case by Lemma
4.7.

Hence, if the proposition fails, then $A_{p}J_{p}=-J_{p}A_{p}$ and
$\sigma_{1}$ is not invariant under $J_{p}$. The following derives a
contradiction.

Let $\\{e_{1},e_{2}\\}$ be an orthonormal basis of $\sigma_{1}$. There exists
a nonzero constant $\mu$ such that $A_{p}e_{1}=\mu e_{2}$ and $A_{p}e_{2}=-\mu
e_{1}$. Rescale $A_{p}$ and replace $e_{2}$ with $-e_{2}$, if necessary, so
that $\mu=1$. If $A_{p}^{*}$ denotes the adjoint of $A_{p}$, then
$A_{p}^{*}=-A_{p}$ on the subspace $\sigma_{1}$.

As $J_{p}$ is orthogonal, $\\{e_{3}=J_{p}e_{1},e_{4}=J_{p}e_{2}\\}$ is an
orthonormal basis of $\sigma_{2}:=J_{p}(\sigma_{1})$. The following
calculations will demonstrate that $\\{e_{1},e_{2},e_{3},e_{4}\\}$ form an
orthonormal $4$-frame. As
$\mathrm{g}_{p}(e_{1},e_{3})=\mathrm{g}_{p}(e_{1},J_{p}e_{1})=0=\mathrm{g}_{p}(e_{2},J_{p}e_{2})=\mathrm{g}_{p}(e_{2},e_{4})$,
it remains to verify the equalities
$\mathrm{g}_{p}(e_{1},e_{4})=0=\mathrm{g}_{p}(e_{2},e_{3}).$ Calculate

$\displaystyle\mathrm{g}_{p}(e_{1},e_{4})=\mathrm{g}_{p}(e_{1},J_{p}e_{2})=\mathrm{g}_{p}(e_{1},J_{p}A_{p}e_{1})=$
$\displaystyle\mathrm{g}_{p}(e_{1},-A_{p}J_{p}e_{1})=\mathrm{g}_{p}(-A_{p}^{*}e_{1},J_{p}e_{1})=$
$\displaystyle\mathrm{g}_{p}(A_{p}e_{1},J_{p}e_{1})=\mathrm{g}_{p}(e_{2},J_{p}e_{1})=$
$\displaystyle\mathrm{g}_{p}(J_{p}e_{2},-e_{1})=-\mathrm{g}_{p}(e_{4},e_{1})$

to conclude that $\mathrm{g}_{p}(e_{1},e_{4})=0$. Finally,

$\mathrm{g}_{p}(e_{2},e_{3})=\mathrm{g}_{p}(J_{p}e_{2},J_{p}e_{3})=\mathrm{g}_{p}(e_{4},-e_{1})=0$

concluding the proof that $\\{e_{1},e_{2},e_{3},e_{4}\\}$ are orthonormal. Let
$\lambda=\sec(\sigma_{1})$ and note that

$\lambda=R(e_{1},e_{2},e_{2},e_{1})=R(J_{p}e_{1},J_{p}e_{2},J_{p}e_{2},J_{p}e_{1})=R(e_{3},e_{4},e_{4},e_{3})=\sec(\sigma_{2}).$

As
$A_{p}(\sigma_{2})=A_{p}J_{p}(\sigma_{1})=-J_{p}A_{p}(\sigma_{1})=-J_{p}(\sigma_{1})=\sigma_{2}$,
the $2$-plane $\sigma_{2}$ is $A_{p}$-invariant. Lemma 4.2 implies that
$\lambda$ is the maximum sectional curvature on the subspace
$V=\sigma_{1}\oplus\sigma_{2}$. By Berger’s curvature inequality ([2, 17]),
$R(e_{1},e_{2},e_{4},e_{3})\leqslant\frac{2}{3}(\lambda-1)$. Consequently,

$\lambda=R(e_{1},e_{2},e_{2},e_{1})=R(e_{1},e_{2},J_{p}e_{2},J_{p}e_{1})=R(e_{1},e_{2},e_{4},e_{3})\leqslant\frac{2}{3}(\lambda-1),$

or equivalently, $\lambda\leqslant-2$, a contradiction. ∎

######  Lemma 4.9.

Assume that $V=\sigma_{1}\oplus\sigma_{2}$ is an orthogonal sum of
$A_{p}$-invariant and $J_{p}$-invariant $2$-plane sections. Let
$v_{i}\in\sigma_{i}$ be unit vectors and $\bar{v}_{i}=A_{p}v_{i}/\mu_{i}$. If
$J_{p}v_{1}=\bar{v}_{1}$, then $J_{p}v_{2}=\bar{v}_{2}$. If
$J_{p}v_{1}=-\bar{v}_{1}$, then $J_{p}v_{2}=-\bar{v}_{2}$. In both cases
$\gamma=R(v_{2},v_{1},\bar{v}_{1},\bar{v}_{2})=1$.

###### Proof.

The assumptions imply that there are constants $c_{1},c_{2}\in\\{-1,1\\}$ such
that $J_{p}v_{i}=c_{i}\bar{v}_{i}$ for $i=1,2$. The first assertion in the
lemma is the equality $c_{1}=c_{2}$ as will now be demonstrated. Note that

$\gamma=R(v_{2},v_{1},\bar{v}_{1},\bar{v}_{2})=R(J_{p}v_{2},J_{p}v_{1},\bar{v}_{1},\bar{v}_{2})=R(c_{2}\bar{v}_{2},c_{1}\bar{v}_{1},\bar{v}_{1},\bar{v}_{2})=c_{1}c_{2}$

where Lemma 4.1 is used in the last equality. By Lemma 4.4, $\gamma>0$ whence
$c_{1}=c_{2}$ and $\gamma=1$. ∎

######  Corollary 4.10.

If $\sigma\subset T_{p}M$ is a $2$-plane section satisfying
$A_{p}(\sigma)=\sigma$, then $J_{p}(\sigma)=\sigma$.

###### Proof.

After possibly rescaling $A_{p}$, there exists an orthonormal basis
$\\{e_{1},e_{2}\\}$ of $\sigma$ satisfying $A_{p}e_{1}=e_{2}$ and
$A_{p}e_{2}=-e_{1}$. If $J_{p}(\sigma)\neq\sigma$ then
$J_{p}(\sigma)\cap\sigma=\\{0\\}$. Letting $e_{3}=J_{p}e_{1}$ and
$e_{4}=J_{p}e_{2}$, the vectors $\\{e_{1},e_{2},e_{3},e_{4}\\}$ span a
$4$-dimensional subspace of $T_{p}M$.

By Proposition 4.8, $A_{p}e_{3}=e_{4}$ and $A_{p}e_{4}=-e_{3}$ since

$A_{p}e_{3}=A_{p}J_{p}e_{1}=J_{p}A_{p}e_{1}=J_{p}e_{2}=e_{4}$

and

$A_{p}e_{4}=A_{p}J_{p}e_{2}=J_{p}A_{p}e_{2}=-J_{p}e_{1}=-e_{3}.$

Let $v_{1}=\frac{e_{1}+e_{4}}{\sqrt{2}}$ and
$v_{2}=\frac{e_{1}-e_{4}}{\sqrt{2}}$ and use the above to calculate
$\bar{v}_{1}=\frac{e_{2}-e_{3}}{\sqrt{2}}$ and
$\bar{v}_{2}=\frac{e_{2}+e_{3}}{\sqrt{2}}$. Verify that
$\sigma_{1}=\operatorname{span}\\{v_{1},\bar{v}_{1}\\}$ and
$\sigma_{2}=\operatorname{span}\\{v_{2},\bar{v}_{2}\\}$ are orthogonal
$A_{p}$-invariant and $J_{p}$-invariant $2$-planes and that
$J_{p}v_{1}=-\bar{v}_{1}$ and $J_{p}v_{2}=\bar{v}_{2}$. This contradicts Lemma
4.9. ∎

###  4.2. Proof of Theorem D when $d=\dim_{\mathbb{R}}(M)\geqslant 6$

######  Lemma 4.11.

For $p\in M$, $A_{p}$ has at most two distinct eigenvalues.

###### Proof.

If not, then there exist three orthogonal $A_{p}$-invariant $2$-planes
$\sigma_{i}$, $i=1,2,3$ and constants $0<\mu_{1}<\mu_{2}<\mu_{3}$ such that
$\|A(w_{i})\|=\mu_{i}$ for each unit vector $w_{i}\in\sigma_{i}$. Let
$\lambda_{i}=\sec(\sigma_{i})$. As $\mu_{1}<\mu_{2}$, Corollary 4.3 implies
that $\lambda_{1}<\lambda_{2}$. By Lemmas 4.5 and 4.9, $\lambda_{2}>4$. As
$\mu_{2}<\mu_{3}$, Corollary 4.3 implies that $\lambda_{2}<\lambda_{3}$. By
Lemmas 4.5 and 4.9, $\lambda_{2}<4$, a contradiction. ∎

######  Lemma 4.12.

If $d=\dim_{\mathbb{R}}(M)\geqslant 6$, then $A_{p}$ has a single eigenvalue
for each $p\in M$.

###### Proof.

If not, Lemma 4.11 implies that there exist constants $0<\mu_{1}<\mu_{2}$ and
$A_{p}$-eigenspaces $E_{1}$ and $E_{2}$ such that $T_{p}M$ is the orthogonal
direct sum $T_{p}M=E_{1}\oplus E_{2}$ and $\|A_{p}(v_{i})\|=\mu_{i}$ for each
unit vector $v_{i}\in E_{i}$, $i=1,2$. As $\dim_{\mathbb{R}}(M)\geqslant 6$,
one of the two eigenspaces $E_{1}$ or $E_{2}$ has real dimension at least
four.

Case I: $\dim_{\mathbb{R}}(E_{1})\geqslant 4$

Choose orthogonal $A_{p}$-invariant $2$-planes $\sigma_{1},\sigma_{2}\subset
E_{1}$ and $\sigma_{3}\subset E_{2}$. Let $\lambda_{i}=\sec(\sigma_{i})$ for
each $i=1,2,3$. As $\mu_{1}<\mu_{2}$, Corollary 4.3 implies that
$\lambda_{1}<\lambda_{3}$ and $\lambda_{2}<\lambda_{3}$. Apply Lemmas 4.5 and
4.9 to the four dimensional subspaces $\sigma_{1}\oplus\sigma_{3}$ and
$\sigma_{2}\oplus\sigma_{3}$ to deduce $\lambda_{1}<4$ and $\lambda_{2}<4$.
Apply Lemmas 4.4 and 4.9 to the four dimensional subspace
$\sigma_{1}\oplus\sigma_{2}$ to deduce $\lambda_{1}+\lambda_{2}=8$, a
contradiction.

Case II: $\dim_{\mathbb{R}}(E_{2})\geqslant 4$

Choose orthogonal $A_{p}$-invariant $2$-planes $\sigma_{1}\subset E_{1}$ and
$\sigma_{2},\sigma_{3}\subset E_{2}$. Let $\lambda_{i}=\sec(\sigma_{i})$ for
each $i=1,2,3$. As $\mu_{1}<\mu_{2}$, Corollary 4.3 implies that
$\lambda_{1}<\lambda_{2}$ and $\lambda_{1}<\lambda_{3}$. Apply Lemmas 4.5 and
4.9 to the four dimensional subspaces $\sigma_{1}\oplus\sigma_{2}$ and
$\sigma_{1}\oplus\sigma_{3}$ to deduce $\lambda_{2}>4$ and $\lambda_{3}>4$.
Applying Lemmas 4.4 and 4.9 to the four dimensional subspace
$\sigma_{2}\oplus\sigma_{3}$ to deduce $\lambda_{2}+\lambda_{3}=8$, a
contradiction. ∎

######  Remark 4.13.

When $\dim_{\mathbb{R}}(M)\geqslant 6$, Theorem D is easily derived from Lemma
4.12 and Theorem C. This approach is taken when $\dim_{\mathbb{R}}=4$ in the
next section.

In the remainder of this section, a more elementary proof is presented for the
case when $\dim_{\mathbb{R}}(M)\geqslant 6$. This alternative proof is based
on the well-known classification [14, 16] of simply-connected Kählerian
manifolds having constant holomorphic curvatures.

######  Corollary 4.14.

A $2$-plane $\sigma\subset T_{p}M$ is holomorphic if and only if
$A_{p}(\sigma)=\sigma$.

###### Proof.

Fix $p\in M$ and let $\sigma\subset T_{p}M$ be a $2$-plane. If
$A_{p}(\sigma)=\sigma$ then $J_{p}(\sigma)=\sigma$ by Corollary 4.10.
Conversely, assume that $J_{p}(\sigma)=\sigma$ and let $v\in\sigma$ be a
nonzero vector. The $2$-plane $\bar{\sigma}=\operatorname{span}\\{v,A_{p}v\\}$
is $A_{p}$-invariant by Lemma 4.12. By Corollary 4.10, $\bar{\sigma}$ is
$J_{p}$-invariant. As $v$ lies in a unique holomorphic $2$-plane,
$\sigma=\bar{\sigma}$, so that $\sigma$ is $A_{p}$-invariant. ∎

######  Corollary 4.15.

If $d=\dim_{\mathbb{R}}(M)\geqslant 6$, then $\lambda(v)=4$ for every unit
vector $v\in SM$.

###### Proof.

Given $v\in S_{p}M$, the $2$-plane
$\sigma_{1}=\operatorname{span}\\{v,A_{p}v\\}$ is $A_{p}$-invariant by Lemma
4.12. As $\dim_{\mathbb{R}}(M)\geqslant 6$, there exist orthogonal
$A_{p}$-invariant $2$-planes $\sigma_{2},\sigma_{3}\subset\sigma_{1}^{\perp}$.
Let $\lambda_{i}=\sec(\sigma_{i})$ for $i=1,2,3$ and note that
$\lambda(v)=\lambda_{1}$.

Applying Lemmas 4.4 and 4.9 to the three four dimensional subspaces
$\sigma_{i}\oplus\sigma_{j}$, $i,j\in\\{1,2,3\\}$ distinct, yields the linear
system

$\lambda_{1}+\lambda_{2}=\lambda_{1}+\lambda_{3}=\lambda_{2}+\lambda_{3}=8,$

whose solution $\lambda_{1}=\lambda_{2}=\lambda_{3}=4$ is unique. ∎

######  Theorem 4.16.

A Kählerian manifold with $\sec\geqslant 1$, real dimension $d\geqslant 6$,
and spherical rank at least $d-2$ is isometric to a globally symmetric
$\mathbb{C}\mathbb{P}^{d/2}$ with holomorphic curvatures equal to $4$.

###### Proof.

It suffices to prove that all holomorphic $2$-planes in $M$ have sectional
curvature equal to four by [14, 16]. Let $p\in M$ and let $\sigma\subset
T_{p}M$ be a holomorphic $2$-plane. Let $v\in\sigma$ be a nonzero vector. By
Corollary 4.14, $\sigma$ is $A_{p}$-invariant, so that
$\sec(\sigma)=\sec(v,A_{p}v)=\lambda(v)$. By Corollary 4.15, $\lambda(v)=4$. ∎

##  5\. Proof of Theorem D in real dimension four

This final section completes the proof of Theorem D, establishing its veracity
when $d=\dim_{\mathbb{R}}(M)=4$. The approach, alluded to in Remark 4.13, is
to appeal to Theorem C. The main step in proving that $M$ satisfies the Rakić
duality principle is to establish the analogue of Lemma 4.12 when $d=4$. The
following lemma, likely well-known, is used for this purpose.

######  Lemma 5.1.

Let $B$ be an open connected subset of a Riemannian manifold $(M,\mathrm{g})$
admitting a pair of orthogonal and totally geodesic foliations
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$. Then $B$ is locally isometric to the
product $\mathcal{F}_{1}\times\mathcal{F}_{2}$.

###### Proof.

If $H=T\mathcal{F}_{1}$ and $V=T\mathcal{F}_{2}$, then the tangent bundle
splits orthogonally $TB=H\oplus V$. By de Rham’s splitting theorem, it
suffices to prove that the distribution $H$ is parallel on $B$. Let
$h,\bar{h}$ denote vector fields tangent to $H$ and let $v,\bar{v}$ denote
vector fields tangent to $V$.

As $H$ is integrable, $0=\mathrm{g}([h,\bar{h}],v),$ implying
$\mathrm{g}(\nabla_{h}\bar{h},v)=\mathrm{g}(\nabla_{\bar{h}}h,v)$. As $H$ is
totally geodesic,
$\mathrm{g}(\nabla_{h}\bar{h},v)=-\mathrm{g}(\nabla_{\bar{h}}h,v)$. Conclude
that

(5.1) $\mathrm{g}(\nabla_{\bar{h}}h,v)=0.$

Similarly, the fact that $V$ is integrable and totally geodesic implies that
$\mathrm{g}(\nabla_{\bar{v}}v,h)=0$. As $H$ and $V$ are orthogonal, this
implies

(5.2) $\mathrm{g}(\nabla_{\bar{v}}h,v)=0.$

By (5.1) and (5.2), $H$ is parallel on $B$, concluding the proof. ∎

Recall from the proof of item (3) of Theorem B that there exists a smooth
section $p\ni M\mapsto A_{p}\in SL(T_{p}M)$.

######  Proposition 5.2.

Assume that $d=\dim_{\mathbb{R}}(M)=4$. Then for each $p\in M$, $A_{p}$ has a
single eigenvalue.

###### Proof.

If not, then there exists a metric ball $B$ in $M$ with the property that for
each $b\in B$, $A_{b}$ has two distinct eigenvalues. For each $b\in B$, there
exist constants $0<\mu_{1}(b)<\mu_{2}(b)$ and orthogonal eigenplanes
$\sigma_{1}(b)$ and $\sigma_{2}(b)$ of $A_{b}$ satisfying
$\|A_{b}(v_{i})\|=\mu_{i}(b)$ for each unit vector $v_{i}\in\sigma_{i}(b)$. As
the $A_{b}$ vary smoothly with $b\in B$, the functions
$\mu_{i}:B\rightarrow\mathbb{R}$ and the orthogonal splitting
$TB=\sigma_{1}\oplus\sigma_{2}$ are both smooth. Define
$\lambda_{i}:B\rightarrow\mathbb{R}$ by $\lambda_{i}(b)=\sec(\sigma_{i}(b))$
for $i=1,2$.

After possibly reducing the radius of $B$, there exist smooth unit vector
fields $v_{1}$ and $v_{2}$ on $B$ tangent to $\sigma_{1}$ and $\sigma_{2}$
respectively. By Corollary 4.10, the two $2$-plane fields $\sigma_{1}$ and
$\sigma_{2}$ are $J$-invariant. Therefore, letting $\bar{v}_{i}=Jv_{i}$, the
smooth orthonormal framing $\\{v_{1},\bar{v}_{1},v_{2},\bar{v}_{2}\\}$ of $TB$
satisfies $\sigma_{i}=\operatorname{span}\\{v_{i},\bar{v}_{i}\\}$ for $i=1,2$.
Define $\gamma:B\rightarrow\mathbb{R}$ by
$\gamma=R(v_{2},v_{1},\bar{v}_{1},\bar{v}_{2})$. Again by Corollary 4.10, the
$A_{b}$-invariant $2$-planes $\sigma_{i}(b)$ are $J_{b}$-invariant and by
Lemma 4.9, $\gamma=1$ on $B$.

Corollary 4.3, implies that $\lambda_{1}(b)<\lambda_{2}(b)$ and Lemma 4.5
implies

(5.3) $\lambda_{1}(b)<4<\lambda_{2}(b)$

for each $b\in B$.

The goal of the following calculations is to show that the orthogonal
distributions $\sigma_{1}$ and $\sigma_{2}$ are integrable and totally
geodesic. As $J$ is parallel,

(5.4)
$\mathrm{g}(\nabla_{X}JY,Z)=\mathrm{g}(J\nabla_{X}Y,Z)=-\mathrm{g}(\nabla_{X}Y,JZ)$

for all smooth vector fields $X,Y,Z$.

Use (5.4) to conclude

(5.5)
$\mathrm{g}(\nabla_{v_{2}}v_{2},\bar{v}_{1})=-\mathrm{g}(\nabla_{v_{2}}\bar{v}_{2},v_{1}).$

Use the differential Bianchi identity,

$0=(\nabla_{v_{2}}R)(v_{1},\bar{v}_{1},v_{1},v_{2})+(\nabla_{v_{1}}R)(\bar{v}_{1},v_{2},v_{1},v_{2})+(\nabla_{\bar{v}_{1}}R)(v_{2},v_{1},v_{1},v_{2})$

to derive

(5.6)
$(\lambda_{1}-1)\mathrm{g}(\nabla_{v_{2}}v_{2},\bar{v}_{1})+3\mathrm{g}(\nabla_{v_{2}}\bar{v}_{2},v_{1})=0.$

Use (5.3), (5.5), and (5.6) to conclude

(5.7)
$\mathrm{g}(\nabla_{v_{2}}v_{2},\bar{v}_{1})=\mathrm{g}(\nabla_{v_{2}}\bar{v}_{2},v_{1})=0.$

Set $w_{1}:=\bar{v}_{1}$ and $\bar{w}_{1}:=Jw_{1}=-v_{1}$. Repeating the above
calculations with $w_{1}$ and $\bar{w}_{1}$ in place of $v_{1}$ and
$\bar{v}_{1}$, respectively, yields the following analogue of (5.7)

(5.8)
$\mathrm{g}(\nabla_{v_{2}}v_{2},\bar{w}_{1})=\mathrm{g}(\nabla_{v_{2}}\bar{v}_{2},w_{1})=0,$

or equivalently,

(5.9)
$\mathrm{g}(\nabla_{v_{2}}v_{2},v_{1})=\mathrm{g}(\nabla_{v_{2}}\bar{v}_{2},\bar{v}_{1})=0.$

Set $w_{2}:=\bar{v}_{2}$ and $\bar{w}_{2}:=Jw_{2}=-v_{2}$. Repeating the above
calculations with $w_{2}$ and $\bar{w}_{2}$ in place of $v_{2}$ and
$\bar{v}_{2}$, respectively, yields the following analogues of (5.7) and (5.9)

(5.10)
$\mathrm{g}(\nabla_{w_{2}}w_{2},\bar{v}_{1})=\mathrm{g}(\nabla_{w_{2}}\bar{w}_{2},v_{1})=0,$

and

(5.11)
$\mathrm{g}(\nabla_{w_{2}}w_{2},v_{1})=\mathrm{g}(\nabla_{w_{2}}\bar{w}_{2},\bar{v}_{1})=0,$

or equivalently,

(5.12)
$\mathrm{g}(\nabla_{\bar{v}_{2}}\bar{v}_{2},\bar{v}_{1})=\mathrm{g}(\nabla_{\bar{v}_{2}}v_{2},v_{1})=0,$

and

(5.13)
$\mathrm{g}(\nabla_{\bar{v}_{2}}\bar{v}_{2},v_{1})=\mathrm{g}(\nabla_{\bar{v}_{2}}v_{2},\bar{v}_{1})=0.$

The $2$-plane field $\sigma_{2}$ is integrable and totally geodesic by (5.7),
(5.9), (5.12), and (5.13).

Switching the roles of the indices $1$ and $2$ in the differential Bianchi
calculation above, yields the following analogue of (5.6)

(5.14)
$(\lambda_{2}-1)\mathrm{g}(\nabla_{v_{1}}v_{1},\bar{v}_{2})+3\mathrm{g}(\nabla_{v_{1}}\bar{v}_{1},v_{2})=0.$

Now, arguing as in the case of the $2$-plane field $\sigma_{2}$, the $2$-plane
field $\sigma_{1}$ is also integrable and totally geodesic. As the tangent
$2$-plane fields $\sigma_{1}$ and $\sigma_{2}$ are orthogonal, integrable, and
totally geodesic, $B$ is locally isometric to a Riemannian product by Lemma
5.1. This contradicts the curvature assumption $\sec\geqslant 1$. ∎

######  Theorem 5.3.

A Kählerian manifold with $\sec\geqslant 1$, real dimension $d=4$, and
spherical rank at least $2$ is isometric to a globally symmetric
$\mathbb{C}\mathbb{P}^{2}$ with holomorphic curvatures equal to $4$.

###### Proof.

It suffices to prove that $M$ satisfies the Rakić duality principle by Theorem
C.

Let $p\in M$ and let $v,w\in S_{p}M$ be a pair of orthonormal vectors. The
Jacobi operator $\mathcal{J}_{v}$ has two eigenspaces, namely the two-
dimensional $1$-eigenspace $E_{v}$ and the one-dimensional
$\lambda(v)$-eigenspace spanned by the vector $A_{p}v$. Similarly, the Jacobi
operator $\mathcal{J}_{w}$ has a two-dimensional $1$-eigenspace $E_{w}$ and a
one-dimensional $\lambda(w)$-eigenspace spanned by $A_{p}w$.

If $w\in E_{v}$, then $v\in E_{w}$ by Lemma 2.1. If $w$ lies in the
$\lambda(v)$-eigenspace of $\mathcal{J}_{v}$, then $w$ is a multiple of
$A_{p}v$. By Proposition 5.2 the $2$-plane
$\sigma:=\operatorname{span}\\{v,w\\}$ is $A_{p}$-invariant, whence
$\lambda(w)=\sec(\sigma)=\lambda(v)$ and $v$ lies in the
$\lambda(w)$-eigenspace of $\mathcal{J}_{w}$. ∎

Together, Theorems 4.16 and 5.3 complete the proof of Theorem D.

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