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# Note on $C^{*}$-algebras associated to boundary actions of hyperbolic
$3$-manifold groups

Shirly Geffen  Shirly Geffen, Mathematisches Institut, Fachbereich Mathematik
und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster,
Germany<EMAIL_ADDRESS>https://shirlygeffen.com/ and  Julian Kranz
Julian Kranz, Institute for Analysis and Numerics, Applied Mathematics
Münster, Faculty of Mathematics and Computer Science, University of Münster,
Einsteinstrasse 62, 48149 Münster, Germany<EMAIL_ADDRESS>https://sites.google.com/view/juliankranz/

###### Abstract.

Using Kirchberg–Phillips’ classification of purely infinite $C^{*}$-algebras
by $K$-theory, we prove that the isomorphism types of crossed product
$C^{*}$-algebras associated to certain hyperbolic $3$-manifold groups acting
on their Gromov boundary only depend on the manifold’s homology. As a result,
we obtain infinitely many pairwise non-isomorphic hyperbolic groups all of
whose associated crossed products are isomorphic. These isomomorphisms are not
of dynamical nature in the sense that they are not induced by isomorphisms of
the underlying groupoids.

###### Key words and phrases:

hyperbolic $3$-manifolds, Crossed products, $K$-theory for $C^{*}$-algebras

###### 2020 Mathematics Subject Classification:

Primary 46L80, 57K32, Secondary 46L35

J. Kranz was supported by the Engineering and Physical Sciences Research
Council [Grant Ref: EP/X026647/1]. S. Geffen was partially supported by the
ERC Advanced Grant 834267 - AMAREC, by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) - Project-ID 427320536 - SFB 1442, as well
as by Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster:
Dynamics-Geometry-Structure.

##  1\. Introduction

The Kirchberg–Phillips classification theorem [Kir94, Phi00] classifies so-
called _UCT Kirchberg $C^{*}$-algebras_ up to isomorphism by topological
$K$-theory. As a consequence, this theorem produces many ‘surprising’
isomorphisms of $C^{*}$-algebras that cannot be constructed by hand, such as
Kirchberg’s isomorphism
$\mathcal{O}_{2}\otimes\mathcal{O}_{2}\cong\mathcal{O}_{2}$. The class of UCT
Kirchberg $C^{*}$-algebras contains many examples such as the (reduced)
crossed product $C^{*}$-algebra $C(\partial G)\rtimes_{r}G$ associated to the
action of a non-elementary torsion-free hyperbolic group $G$ on its Gromov-
boundary [AD97, Tu99]. In fact, crossed products of arbitrary minimal amenable
topologically free actions $G\curvearrowright X$ of acylindrically hyperbolic
groups belong to this class [GGKN23]. Although this large class of
$C^{*}$-algebras is _in principle_ classifiable by $K$-theory, explicit
$K$-theory computations remain difficult in general. For the specific example
of the action of a torsion-free hyperbolic group $G$ on its boundary, Emerson
and Meyer [EM06] relate the $K$-theory of $C(\partial G)\rtimes_{r}G$ to the
$K$-theory and $K$-homology of the classifying space $BG$. In this short note,
we use their result to obtain explicit computations for fundamental groups of
hyperbolic $3$-manifolds.

######  Theorem A.

Let $d\geq 0$ be a fixed integer and let $G$ be the fundamental group of a
closed connected orientable hyperbolic $3$-manifold $M$ with
$H_{1}(M)\cong\mathbb{Z}^{d}$. Then there are isomorphisms

$K_{0}(C(\partial G)\rtimes_{r}G)\cong K_{1}(C(\partial
G)\rtimes_{r}G)\cong\mathbb{Z}^{2d+2}$

that identify the class of the unit $[1]\in K_{0}(C(\partial G)\rtimes_{r}G)$
with $(1,0)\in\mathbb{Z}\oplus\mathbb{Z}^{2d+1}$.

Brock–Dunfield’s work [BD15] shows that there exists a rich supply of groups
that satisfy the assumptions in Theorem A. Thus, we obtain examples of
isomorphisms of crossed product $C^{*}$-algebras:

######  Corollary .

There are infinitely many pairwise non-isomorphic hyperbolic groups
$G_{i},\leavevmode\nobreak\ i\in\mathbb{N}$ with pairwise isomorphic crossed
products $C(\partial G_{i})\rtimes_{r}G_{i},\leavevmode\nobreak\
i\in\mathbb{N}$.

These isomorphisms are ‘surprising’ in the sense that they cannot be obtained
from an isomorphism of the underlying groupoids.

######  Proposition .

Let $G_{i}$ and $G_{j}$ be two non-isomorphic groups as in Theorem A. Then
there is no isomorphism $C(\partial G_{i})\rtimes_{r}G_{i}\cong C(\partial
G_{j})\rtimes_{r}G_{j}$ that restricts to an isomorphism $C(\partial
G_{i})\cong C(\partial G_{j})$. In particular, $C(\partial
G_{i})\rtimes_{r}G_{i}$ contains infinitely many pairwise non-conjugate Cartan
subalgebras with spectrum $S^{2}$.

### Acknowledgments

This note arose from discussions with Johannes Ebert and is mostly based on
his ideas. We are indebted to him for sharing his strategy to combine the
Baum–Connes conjecture with the Atiyah–Hirzebruch spectral sequence, for
educating us about hyperbolic $3$-manifolds, for guiding us through the
computations, and for encouraging us to use his ideas and publish this note.
We are moreover grateful to Jakub Curda for pointing out the reference [Mat02]
which drastically simplified the proof.

##  2\. Isomorphism types of crossed products

Let $M$ be a closed connected orientable hyperbolic $3$-manifold. Then the
universal cover $\widetilde{M}$ of $M$ is isometric to the $3$-dimensional
hyperbolic space $\mathbb{H}^{3}$ by the Killing–Hopf theorem [Lee18,
Corollary 12.5]. Let $G=\pi_{1}(M)$ be the fundamental group of $M$.
Considering the natural (free) action $G\curvearrowright\widetilde{M}$ by Deck
transformations, and the fact that $\widetilde{M}$ is contractible, the orbit
space $\widetilde{M}/G\cong M$ is a finite model for the classifying space
$BG$. It follows that $G$ is finitely generated and torsion-free [Bro82,
§VIII, Exercise 4.1 and Corollary 2.5]. Moreover, since the action
$G\curvearrowright\widetilde{M}$ is proper and co-compact, $G$ is quasi-
isometric to $\mathbb{H}^{3}$ by the Milnor-Švarc lemma [BH99, §I.8,
Proposition 8.9]. In particular, $G$ is a Gromov-hyperbolic group with Gromov
boundary $\partial G\cong\partial\mathbb{H}^{3}\cong S^{2}$ [BH99, §III H,
Theorem 1.9 and Theorem 3.9].

We refer the reader to [CELY17] for the definition of crossed product
$C^{*}$-algebras, topological $K$-theory for $C^{*}$-algebras, and the
Baum–Connes conjecture. For a unital $C^{*}$-algebra $A$, we denote by
$[1_{A}]\in K_{0}(A)$ the class of its unit (or equivalently the class of the
rank-one $A$-module $[A]$).

For a connected space $X$ and a choice of base point $x_{0}\in X$, we denote
by $[x_{0}]\in H_{0}(X)$ the image of the canonical generator
$1\in\mathbb{Z}=H_{0}(\\{x_{0}\\})$ under the natural map
$H_{0}(\\{x_{0}\\})\to H_{0}(X)$ and by $[1_{X}]\in K_{0}(X)$ the image of the
canonical generator $1\in\mathbb{Z}=K_{0}(\\{x_{0}\\})$ under the natural map
$K_{0}(\\{x_{0}\\})\to K_{0}(X)$.

The following computations of $K$-theory and $K$-homology of $3$-manifolds
using their cohomology and homology groups is an application of the Atiyah-
Hirzebruch spectral sequence. The proof is given in [Mat02, Proposition 2.1],
but we include a brief introduction to spectral sequences and more detailed
computations for non-experts in Appendix A.

######  Lemma 2.1.

Let $M$ be a closed connected orientable $3$-manifold. Then there are
isomorphisms

(1) $\displaystyle K_{0}(M)$ $\displaystyle\cong H_{0}(M)\oplus H_{2}(M),$ (2)
$\displaystyle K_{1}(M)$ $\displaystyle\cong H_{1}(M)\oplus H_{3}(M),$ (3)
$\displaystyle K^{0}(M)$ $\displaystyle\cong H^{0}(M)\oplus H^{2}(M),$ (4)
$\displaystyle K^{1}(M)$ $\displaystyle\cong H^{1}(M)\oplus H^{3}(M),$

that identify $[1_{M}]\in K_{0}(M)$ with $[x_{0}]\in H_{0}(M)$.

###### Proof.

The isomorphisms (1) and (2) identifying $[1_{M}]$ with $[x_{0}]$ are obtained
in [Mat02, Proposition 2.1]. By Poincaré duality, we have $H^{*}(M)\cong
H_{3-*}(M)$. Moreover, by Kasparov’s $K$-theoretic Poincaré duality [Kas88,
Theorem 4.10]111In order to apply [Kas88, Theorem 4.10], recall that as a
closed oriented $3$-manifold, $M$ admits a $\mathrm{spin}^{c}$-structure and
that any $\mathrm{spin}^{c}$-structure induces a $\mathbb{Z}/2$-graded Morita-
equivalence $C_{\tau}(M)\sim C(M)\hat{\otimes}\operatorname{\mathrm{Cl}}_{3}$,
where $C_{\tau}(M)$ denotes the section algebra of the Clifford bundle of
$T^{*}M$ and $\operatorname{\mathrm{Cl}}_{3}$ the third complex Clifford
algebra., we have an isomorphism $K^{*}(M)\cong K_{3-*}(M)$. Together with
Bott periodicity, this proves (3) and (4). ∎

###### Proof of Theorem A.

Using Poincaré duality and
$H^{1}(M)=\operatorname{Hom}_{\mathbb{Z}}(H_{1}(M),\mathbb{Z})$, the
assumptions in Theorem A imply the following isomorphisms:

$\displaystyle H_{0}(M)$ $\displaystyle\cong H^{3}(M)\cong H^{0}(M)\cong
H_{3}(M)\cong\mathbb{Z},$ $\displaystyle H_{1}(M)$ $\displaystyle\cong
H^{2}(M)\cong H^{1}(M)\cong H_{2}(M)\cong\mathbb{Z}^{d}.$

Combining with Lemma 2.1, these allow to compute

(5) $K_{0}(M)\cong K_{1}(M)\cong K^{0}(M)\cong K^{1}(M)\cong\mathbb{Z}^{d+1}$

and identify

(6) $K_{0}(M)\ni[1_{M}]\mapsto(1,0)\in\mathbb{Z}\oplus\mathbb{Z}^{d}.$

Note that $M$ has zero Euler characteristic by Poincaré duality. Moreover, $G$
satisfies the Baum–Connes conjecture with coefficients by [Laf12]. Thus,
[EM06, Theorem 1] applied to the action $G\curvearrowright\widetilde{M}$
produces short exact sequences

(7) $\displaystyle\begin{aligned} 0\to
K_{0}(C_{r}^{*}(G))\xrightarrow{u_{*}}K_{0}(C(\partial G)\rtimes_{r}G)\to
K^{1}(M)\to 0,\\\ 0\to K_{1}(C_{r}^{*}(G))\xrightarrow{u_{*}}K_{1}(C(\partial
G)\rtimes_{r}G)\to K^{0}(M)\to 0,\end{aligned}$

where the map $u_{*}$ is induced by the canonical inclusion $u\colon
C^{*}_{r}(G)\to C(\partial G)\rtimes_{r}G$. Recall that $M$ is a model for
$BG$. Thus, the Baum–Connes assembly map

(8) $\mu^{G}_{*}\colon K_{*}(M)\xrightarrow{\cong}K_{*}(C^{*}_{r}(G))$

is an isomorphism. By applying functoriality of the assembly map to the group
homomorphism $\\{e\\}\to G$, we moreover conclude that

(9) $\mu^{G}_{0}([1_{M}])=[1_{C^{*}_{r}(G)}]\in K_{0}(C^{*}_{r}(G)).$

Indeed, this follows by applying commutativity of the diagram

$\textstyle{K_{0}(\operatorname{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\\{e\\}}_{0}}$$\textstyle{K_{0}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{G}_{0}}$$\textstyle{K_{0}(C^{*}_{r}(G))}$

to the element $[1_{\operatorname{pt}}]\in K_{0}(\operatorname{pt})$.

Since $K^{*}(M)\cong\mathbb{Z}^{d+1}$, the short exact sequences in (7) split.
Using (8) at the second step and (5) at the third step, we get

$K_{0}(C(\partial G)\rtimes_{r}G)\cong K_{0}(C^{*}_{r}(G))\oplus K^{1}(M)\cong
K_{0}(M)\oplus K^{1}(M)\cong\mathbb{Z}^{2d+2}.$

By (6) and (9), this isomorphism identifies the class of the unit
$[1_{C(\partial G)\rtimes_{r}G}]=u_{*}([1_{C^{*}_{r}(G)}])\in{K_{0}(C(\partial
G)\rtimes_{r}G)}$ with $(1,0)\in\mathbb{Z}\oplus\mathbb{Z}^{2d+1}$.
Analogously, we obtain an isomorphism

$K_{1}(C(\partial G)\rtimes_{r}G)\cong K_{1}(C^{*}_{r}(G))\oplus K^{0}(M)\cong
K_{1}(M)\oplus K^{0}(M)\cong\mathbb{Z}^{2d+2}.$

∎

###### Proof of Section 1.

By [BD15, Theorem 2.1], there are infinitely many pairwise non-isometric
closed connected orientable222Note that although not stated in the theorem,
the proof of [BD15, Theorem 2.1] indeed produces orientable manifolds.
hyperbolic $3$-manifolds $M_{i},\leavevmode\nobreak\ i\in\mathbb{N}$
satisfying $H_{1}(M_{i})\cong\mathbb{Z}^{d}$. In particular, the fundamental
groups $G_{i}\coloneqq\pi_{1}(M_{i}),\leavevmode\nobreak\ i\in\mathbb{N}$ are
pairwise non-isomorphic by Mostow’s rigidity theorem [Mos68].

The crossed products $A_{i}\coloneqq C(\partial G_{i})\rtimes_{r}G_{i}$ are
unital simple separable nuclear purely infinite $C^{*}$-algebras by [AD97,
Proposition 3.2] that satisfy the UCT by [Tu99, Proposition 10.7]. Fix
$i,j\in\mathbb{N}$. Theorem A implies that there are isomorphisms
$K_{0}(A_{i})\cong K_{0}(A_{j})$ and $K_{1}(A_{i})\cong K_{1}(A_{j})$ that
identify $[1_{A_{i}}]$ with $[1_{A_{j}}]$. Thus, the Kirchberg–Phillips
classification theorem [Kir94, Phi00] implies that $A_{i}$ and $A_{j}$ are
isomorphic. ∎

######  Remark .

We do not know, whether the reduced group $C^{*}$-algebras
$C^{*}_{r}(G_{i}),\leavevmode\nobreak\ i\in\mathbb{N}$ are pairwise
isomorphic, although $K$-theory does not distinguish them. To the best of our
knowledge, it is an open problem whether the reduced group $C^{*}$-algebra of
a general torsion-free discrete group $G$ recovers the group. However, results
of this type do exist for certain classes of groups (see [ER18] for instance).

As a preparation for the proof of Section 1, recall that the _transformation
groupoid_ $X\rtimes G$ associated to a group action $G\curvearrowright X$ is
the topological groupoid whose object space is given by $(X\rtimes
G)^{(0)}\coloneqq X$ and whose morphism space is given by $X\rtimes G\coloneqq
X\times G$, with range, source, and composition maps $r,s,\cdot$ determined by
the formulas

$r(x,g)=x,\quad s(x,g)=g^{-1}x,\quad(x,g)\cdot(g^{-1}x,h)=(x,gh),\quad x\in
X,g,h\in G.$

The _topological full group_ $[[\mathcal{G}]]$ of a topological groupoid
$\mathcal{G}$ is the group of all global bisections, that is, open subsets
$U\subseteq\mathcal{G}$ of the arrow space such that the restrictions
$r|_{U}\colon U\to\mathcal{G}^{(0)}$ and $s|_{U}\colon U\to\mathcal{G}^{(0)}$
are homeomorphisms333Usually, topological full groups are only defined and
interesting for groupoids with totally disconnected object space, see
[Mat12].. Note that the topological full group of a topological groupoid is an
isomorphism invariant.

######  Lemma 2.2.

Let $X$ be a connected space and $G\curvearrowright X$ an action of a discrete
group. Then the natural map

$G\to[[X\rtimes G]],\quad g\mapsto X\times\\{g\\}$

is a group isomorphism.

###### Proof.

If $U\in[[X\rtimes G]]$ is a global bisection, then $U$ is homeomorphic to $X$
and therefore connected. In particular, the projection of $U\subseteq X\times
G$ to $G$ is constant. ∎

###### Proof of Section 1.

Assume by contradiction that there is an isomorphism

$C(\partial G_{i})\rtimes_{r}G_{i}\cong C(\partial G_{j})\rtimes_{r}G_{j}$

that restricts to an isomorphism $C(\partial G_{i})\cong C(\partial G_{j})$.
Note that the action $G_{i}\curvearrowright\partial G_{i}$ is topologically
free since $G_{i}$ is torsion-free. Thus, $C(\partial G_{i})\subseteq
C(\partial G_{i})\rtimes_{r}G_{i}$ is a Cartan subalgebra (see [Ren08, §6.1]).
By Renault’s reconstruction theorem [Ren08, Theorem 5.9], the isomorphism
$C(\partial G_{i})\rtimes_{r}G_{i}\cong C(\partial G_{j})\rtimes_{r}G_{j}$ is
induced by an isomorphism $\partial G_{i}\rtimes G_{i}\cong\partial
G_{j}\rtimes G_{j}$ of the underlying transformation groupoids. In particular,
their associated topological full groups are isomorphic. By Lemma 2.2, this
implies that $G_{i}\cong G_{j}$, which is a contradiction. ∎

##  3\. $K$-theory with coefficients

We end this note with a variant of Theorem A and Section 1 for $3$-manifolds
with torsion in their first homology. Recall that for an Abelian group $A$ and
a $C^{*}$-algebra $B$, the $K$-theory of $B$ with coefficients in $A$ is
defined as $K_{*}(B;A)\coloneqq K_{*}(B\otimes\mathcal{A})$ where
$\mathcal{A}$ is any $C^{*}$-algebra satisfying the UCT such that
$K_{0}(\mathcal{A})\cong A$ and $K_{1}(\mathcal{A})=0$. We refer to [RS87] for
the UCT and Künneth theorem for $C^{*}$-algebras.

######  Theorem 1.

Let $G$ be the fundamental group of a closed connected orientable hyperbolic
$3$-manifold. Let $\mathbb{F}$ be a field. Then there are isomorphisms

$K_{0}(C(\partial G)\rtimes_{r}G;\mathbb{F})\cong K_{1}(C(\partial
G)\rtimes_{r}G;\mathbb{F})\cong\mathbb{F}^{2}\oplus H_{1}(M;\mathbb{F})\oplus
H^{1}(M;\mathbb{F})$

which identify the class $[1]\in K_{0}(C(\partial G)\rtimes_{r}G;\mathbb{F})$
with $(1,0)\in\mathbb{F}\oplus(\mathbb{F}\oplus H_{1}(M;\mathbb{F})\oplus
H^{1}(M;\mathbb{F}))$.

###### Proof.

Since the proof follows the same strategy as the proof of Theorem A, we only
explain the necessary modifications. Let $\mathcal{A}$ be a $C^{*}$-algebra
satisfying the UCT such that $K_{0}(\mathcal{A})\cong\mathbb{F}$ and
$K_{1}(\mathcal{A})\cong 0$.

The $K$-homology of $M$ can be computed using a version of [Mat02, Proposition
2.1] with coefficients in $\mathbb{F}$, which in turn can be proven in exactly
the same way as the integral version (using the Atiyah–Hirzebruch spectral
sequence for $K$-homology with coefficients in $\mathbb{F}$). Note that the
exact sequences (7) are obtained in [EM06] as long exact sequences in
$K$-theory associated to the short exact sequence

$0\to C_{0}(X)\rtimes_{r}G\to C(\overline{X})\rtimes_{r}G\to C(\partial
G)\rtimes_{r}G\to 0,$

where $X=\widetilde{M}$ is the universal cover of $M$ and $\overline{X}$ is
its natural compactification. Tensoring this sequence with $\mathcal{A}$ and
applying $K$-theory yields a new six-term exact sequence which again splits
into two short exact sequences by the Künneth theorem (note that $\mathcal{A}$
satisfies the Künneth theorem since it satisfies the UCT). In this way we
obtain the appropriate analogues of the Emerson–Meyer exact sequences with
coefficients in $\mathbb{F}$,

$\displaystyle 0\to
K_{0}(C_{r}^{*}(G);\mathbb{F})\xrightarrow{u_{*}}K_{0}(C(\partial
G)\rtimes_{r}G;\mathbb{F})\to K^{1}(M;\mathbb{F})\to 0,$ $\displaystyle 0\to
K_{1}(C_{r}^{*}(G);\mathbb{F})\xrightarrow{u_{*}}K_{1}(C(\partial
G)\rtimes_{r}G;\mathbb{F})\to K^{0}(M;\mathbb{F})\to 0,$

which split since $\mathbb{F}$ is a field. The left and right hand terms of
these sequences are now computed in the same way as before, where Kasparov’s
Poincaré duality theorem [Kas88, Theorem 4.9] and the Baum–Connes isomorphism
[Laf12] have to be taken with coefficients in $\mathcal{A}$. ∎

######  Corollary .

Let $\mathbb{F}$ be a field and let $\mathcal{A}$ be a unital simple separable
nuclear $C^{*}$-algebra satisfying the UCT such that
$K_{0}(\mathcal{A})\cong\mathbb{F}$ and $K_{1}(\mathcal{A})\cong 0$. For
instance, let $\mathcal{A}$ be the universal UHF algebra $\mathcal{Q}$ for
$\mathbb{F}=\mathbb{Q}$, or let $\mathcal{A}$ be the Cuntz algebra
$\mathcal{O}_{p+1}$ for $\mathbb{F}=\mathbb{F}_{p}$. Let $M$ be a finitely
generated Abelian group. Then there are infinitely many pairwise non-
isomorphic hyperbolic groups $G_{i},\leavevmode\nobreak\ i\in\mathbb{N}$ with
$H_{1}(G_{i})\cong M$ such that the stabilized crossed products

$\mathcal{A}\otimes(C(\partial G_{i})\rtimes_{r}G_{i}),\leavevmode\nobreak\
i\in\mathbb{N}$

are all isomorphic.

###### Proof.

Note that the assumptions on $\mathcal{A}$ guarantee that the stabilized
crossed products in the statement are UCT Kirchberg algebras, see [KR00,
Proposition 4.5] and [BO08, Proposition 10.1.7]. Thus, the corollary can be
proven in the same way as Section 1, using [BD15] and using 1 instead of
Theorem A. ∎

##  Appendix A Spectral sequences

In this appendix, we give more details on the computation $K$-theory and
$K$-homology (with integer coefficients) of a closed, connected, orientable,
3-dimensional, hyperbolic manifold in terms of homology and cohomology, see
[Mat02, Proposition 2.1] and Lemma 2.1. This is done using the Atiyah-
Hirzebruch spectral sequence.

We recall that a cohomological spectral sequence (in the category of Abelian
groups) is given by a collection $\\{E_{i},d_{i}\\}_{i\in\mathbb{N}}$, where,
for each $i$, the page $E_{i}$ consists of a grid of Abelian groups, denoted
$(E_{i}^{p,q})_{p,q\in\mathbb{Z}}$, equipped with maps $d_{i}^{p,q}\colon
E_{i}^{p,q}\to E_{i}^{p+i,q-i+1}$, called differentials, which satisfy
$d_{i}^{p,q}\circ d_{i}^{p-i,q+i-1}=0$, or, in other words,
$\mathrm{Im}(d_{i}^{p-i,q+i-1})\subseteq\mathrm{Ker}(d_{i}^{p,q})$. Moreover,
the “next page” $E_{i+1}$ is given by the homology of the page $E_{i}$. That
is,

$E_{i+1}^{p,q}\cong\frac{\mathrm{Ker}(d_{i}^{p,q})}{\mathrm{Im}(d_{i}^{p-i,q+i-1})}.$

The spectral sequence $\\{E_{i},d_{i}\\}_{i\in\mathbb{N}}$ is called
degenerate at $i_{0}$ if the differentials $(d_{i}^{p,q})_{p,q\in\mathbb{Z}}$
vanish for all $i\geq i_{0}$. In this case, the limiting term can be defined
as $E_{\infty}^{p,q}=E_{i}^{p,q}$, for $p,q\in\mathbb{Z}$, and some $i\geq
i_{0}$. We will only deal with degenerate spectral sequences, although the
limiting term can be defined more generally.

Assume that $(A^{n})_{n\in\mathbb{Z}}$ is a collection of Abelian groups, so
that each $A^{n}$ admits a Hausdorff grading, which we write as

$A^{n}\coloneqq G^{0}A^{n}\supseteq G^{1}A^{n}\supseteq
G^{2}A^{n}\supseteq\ldots$

with $\bigcap_{p=0}^{\infty}G^{p}A^{n}=0$. It is said that a (degenerate)
spectral sequence $\\{E_{i},d_{i}\\}_{i\in\mathbb{N}}$ converges to
$(A^{n})_{n\in\mathbb{Z}}$ if for every $p\geq 0$ and $q\in\mathbb{Z}$ we have
$E_{\infty}^{p,q}\cong\frac{G^{p}A^{p+q}}{G^{p+1}A^{p+q}}$. We will use the
notation $E_{r}^{p,q}\Rightarrow A^{p+q}$ for convergence.

######  Lemma.

Let $M$ be a closed connected orientable 3-manifold. We have $K^{0}(M)\cong
H^{0}(M)\oplus H^{2}(M)$ and $K^{1}(M)\cong H^{1}(M)\oplus H^{3}(M)$.

###### Proof.

The second page of the Atiyah-Hirzebruch spectral sequence for $K$-theory
(which applies more generally in the setting of finite CW-complexes) is given
by

$E_{2}^{p,q}=H^{p}(M,K^{q}(\mathrm{pt})).$

We claim that the differentials $d_{2}^{p,q}$ vanish. Indeed, the $K$-theory
of a point is

$K^{q}(\mathrm{pt})=\begin{cases}\mathbb{Z},&\text{if q is even}\\\
0,&\text{otherwise}\end{cases}.$

Moreover, under our assumptions on $M$, we have $H^{n}(M)=0$ for all
$n\notin\\{0,1,2,3\\}$ (this follows, for example, from Poincaré duality and
the fact that cohomology groups are defined to be zero for negative degrees).

Putting these facts together, we can visualize the second page of the Atiyah-
Hirzebruch spectral sequence as follows:

$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\cdots}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\cdots}$$\textstyle{\cdots}$$\textstyle{0}$$\textstyle{H^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{2}}$$\textstyle{H^{1}(M)}$$\textstyle{H^{2}(M)}$$\textstyle{H^{3}(M)}$$\textstyle{0}$$\textstyle{\cdots}$$\textstyle{\cdots}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{2}}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\cdots}$$\textstyle{\cdots}$$\textstyle{0}$$\textstyle{H^{0}(M)}$$\textstyle{H^{1}(M)}$$\textstyle{H^{2}(M)}$$\textstyle{H^{3}(M)}$$\textstyle{0}$$\textstyle{\cdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$

It is immediately visible that the differentials $d_{2}$ either have vanishing
range or vanishing domain, thus must all be the zero map. The third page in
the Atiyah-Hirzebruch spectral sequence is now concluded to be identical (as a
grid) to the second page (since $E_{3}$ is defined to be the homology of
$E_{2}$). We will check that the differentials $d_{3}$ vanish as well. In the
following figure we draw the only (recurring) differentials which could
potentially be non-zero:

$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\cdots}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\cdots}$$\textstyle{\cdots}$$\textstyle{0}$$\textstyle{H^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{3}}$$\textstyle{H^{1}(M)}$$\textstyle{H^{2}(M)}$$\textstyle{H^{3}(M)}$$\textstyle{0}$$\textstyle{\cdots}$$\textstyle{\cdots}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\cdots}$$\textstyle{\cdots}$$\textstyle{0}$$\textstyle{H^{0}(M)}$$\textstyle{H^{1}(M)}$$\textstyle{H^{2}(M)}$$\textstyle{H^{3}(M)}$$\textstyle{0}$$\textstyle{\cdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$

Applying the Atiyah-Hirzebruch spectral sequence to a one-point space, we
obtain exactly the same second and third page as obtained above for the
manifold $M$, except $H^{n}(\mathrm{pt})=0$ for all $n\neq 0$.

Consider the natural maps $\mathrm{pt}\to M\to\mathrm{pt}$. Since the Atiyah-
Hirzebruch spectral sequence is functorial, we have a retract
$E_{3}(\mathrm{pt})\to E_{3}(M)\to E_{3}(\mathrm{pt})$, which is interpreted
as the following commutative diagram.

$\textstyle{H^{0}(\mathrm{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{3}}$$\textstyle{H^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{3}}$$\textstyle{H^{0}(\mathrm{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{3}}$$\textstyle{H^{3}(\mathrm{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}(\mathrm{pt})}$

As $H^{3}(\mathrm{pt})=0$, we have that the differential maps on the left-hand
side and right-hand side must vanish. The composition
$\textstyle{H^{0}(\mathrm{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathrm{pt})}$
is the identity map. By our assumptions on $M$ (and Poincaré duality), we have
$H^{0}(M)\cong H^{3}(M)\cong H^{0}(\mathrm{pt})\cong\mathbb{Z}$, which forces
the maps
$\textstyle{H^{0}(\mathrm{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(M)}$
and
$\textstyle{H^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathrm{pt})}$
to be isomorphisms. Rewriting the diagram above, we have

$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{3}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{3}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{3}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

which now clearly forces the map $d_{3}\colon H^{0}(M)\to H^{3}(M)$ to be the
zero map, as we wanted to show.

Observe that for $n\geq 3$, $E_{n}$ is identical to $E_{2}$ as a grid, and it
is easy to see that the differentials $d_{n}$ vanish as well for all $n\geq
4$. We conclude that the Atiyah-Hirzebruch spectral sequence is degenerate at
$i_{0}=2$, and so $E_{\infty}^{p,q}=E_{2}^{p,q}$, for all $p,q\in\mathbb{Z}$.

Set $A^{n}\coloneqq K^{n}(M)$, for $n\in\mathbb{Z}$. By Atiyah–Hirzebruch,
there exists a Hausdorff grading $A^{n}=G^{0}A^{n}\supseteq
G^{1}A^{n}\supseteq G^{3}A^{n}\supseteq\ldots$, for every $n\in\mathbb{Z}$
(the concrete description of the grading will follow from the computations
below), such that $E_{2}^{p,q}\Rightarrow K^{p+q}(M)$.

For $p,q\in\mathbb{Z}$ with $p+q=0$ we get the following short exact
sequences:

|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{1}A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{0}A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\infty}^{0,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{2}A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{1}A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\infty}^{1,-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{3}A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{2}A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\infty}^{2,-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{4}A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{3}A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\infty}^{3,-3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\vdots}$

The first short exact sequence reduces to

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{G^{1}A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
80.4625pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
80.4625pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
141.04028pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
141.04028pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
201.5347pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
201.5347pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Since $H^{0}(M)\cong\mathbb{Z}$ is torsion free, we conclude that
$K^{0}(M)\cong H^{0}(M)\oplus G^{1}A^{0}$.

If $n$ is odd, we have $E_{\infty}^{n,-n}=0$, so $G^{1}A_{0}\cong G^{2}A^{0}$
and $G^{3}A^{0}\cong G^{4}A^{0}$. Note that $E_{\infty}^{n,-n}=0$ whenever
$n\geq 4$. Thus, $G^{n}A^{0}\cong G^{3}A^{0}$ for all $n\geq 4$. Since the
grading is Hausdorff, it follows that $G^{n}A^{0}=0$ for all $n\geq 3$.
Finally, $E_{\infty}^{2,-2}\cong H^{2}(M)$. Working inductively with the short
exact sequences above, we conclude that

$K^{0}(M)\cong H^{0}(M)\oplus G^{1}A^{0}\cong H^{0}(M)\oplus G^{2}A^{0}\cong
H^{0}(M)\oplus H^{2}(M).$

Next, we compute $K^{1}(M)$ using $E_{2}^{p,q}\Rightarrow K^{p+q}(M)$. For
$p,q\in\mathbb{Z}$ with $p+q=1$ we get the following short exact sequences

|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{1}A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{0}A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\infty}^{0,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{2}A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{1}A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\infty}^{1,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{3}A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{2}A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\infty}^{2,-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{4}A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{3}A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\infty}^{3,-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\vdots}$

Since $E_{\infty}^{0,1}\cong E_{\infty}^{2,-1}\cong 0$, the first short exact
sequence implies that $K^{1}(M)\cong G^{1}A^{1}$, and the third short-exact
sequence implies $G^{2}A^{1}\cong G^{3}A^{1}$. Since $E_{\infty}^{1,0}\cong
H^{1}(M)\cong\mathrm{Hom}(M,\mathbb{Z})$ is torsion-free and finitely
generated, we have that the second short exact sequence splits, and
$G^{1}A^{1}\cong H^{1}(M)\oplus G^{2}A^{1}\cong H^{1}(M)\oplus G^{3}A^{1}$.

Finally, $E_{\infty}^{3,-2}\cong H^{3}(M)\cong H_{0}(M)\cong\mathbb{Z}$ by
Poincaré duality. Thus, the last short exact sequence splits as well, and we
have $G^{3}A^{1}\cong H^{3}(M)\oplus G^{4}A^{1}$. However, noticing that
$E_{\infty}^{n,m}=0$ for all $n\geq 4$ and all $m\in\mathbb{Z}$, we see that
$G^{n}A^{1}\cong G^{4}A^{1}$ for all $n\geq 4$. Since the grading is
Hausdorff, it follows that $G^{4}A^{1}=0$, and so $G^{3}A^{1}\cong H^{3}(M)$.
Combining the above,

$K^{1}(M)\cong G^{1}A^{1}\cong H^{1}(M)\oplus G^{2}A^{1}\cong H^{1}(M)\oplus
G^{3}A^{1}\cong H^{1}(M)\oplus H^{3}(M).$

∎

In a similar manner, one can apply the homological version of the Atiyah-
Hirzebruch spectral sequence in order to compute the $K$-homology of $M$,
$K_{*}(M)$. Alternatively, one can use Kasparov’s $K$-theoretic Poincaré
duality and Bott periodicity (as explained in the proof of Lemma 2.1) to
conclude that $K_{0}(M)\cong H_{0}(M)\oplus H_{2}(M)$ and $K_{1}(M)\cong
H_{1}(M)\oplus H_{3}(M)$.

## References

  * [AD97] Claire Anantharaman-Delaroche.  Purely infinite $C^{*}$-algebras arising from dynamical systems.  Bull. Soc. Math. France, 125(2):199–225, 1997.
  * [BD15] Jeffrey F. Brock and Nathan M. Dunfield.  Injectivity radii of hyperbolic integer homology 3-spheres.  Geom. Topol., 19(1):497–523, 2015.
  * [BH99] Martin R. Bridson and André Haefliger.  Metric spaces of non-positive curvature, volume 319 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 1999.
  * [BO08] Nathanial P. Brown and Narutaka Ozawa.  $C^{*}$-algebras and finite-dimensional approximations, volume 88 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2008.
  * [Bro82] Kenneth S. Brown.  Cohomology of groups, volume 87 of Graduate Texts in Mathematics.  Springer-Verlag, New York-Berlin, 1982.
  * [CELY17] Joachim Cuntz, Siegfried Echterhoff, Xin Li, and Guoliang Yu.  $K$-theory for group $C^{*}$-algebras and semigroup $C^{*}$-algebras, volume 47 of Oberwolfach Seminars.  Birkhäuser/Springer, Cham, 2017.
  * [EM06] Heath Emerson and Ralf Meyer.  Euler characteristics and Gysin sequences for group actions on boundaries.  Math. Ann., 334(4):853–904, 2006.
  * [ER18] Caleb Eckhardt and Sven Raum.  $C^{*}$-superrigidity of 2-step nilpotent groups.  Advances in Mathematics, 338:175–195, 2018.
  * [GGKN23] Eusebio Gardella, Shirly Geffen, Julian Kranz, and Petr Naryshkin.  Classifiability of crossed products by nonamenable groups.  J. Reine Angew. Math., 797:285–312, 2023.
  * [Kas88] Gennadi G. Kasparov.  Equivariant $KK$-theory and the Novikov conjecture.  Invent. Math., 91(1):147–201, 1988.
  * [Kir94] Eberhard Kirchberg.  The classification of purely infinite $C^{*}$-algebras using Kasparov’s theory.  Preprint, 1994.
  * [KR00] Eberhard Kirchberg and Mikael Rørdam.  Non-simple purely infinite $C^{\ast}$-algebras.  Amer. J. Math., 122(3):637–666, 2000.
  * [Laf12] Vincent Lafforgue.  La conjecture de Baum-Connes à coefficients pour les groupes hyperboliques.  J. Noncommut. Geom., 6(1):1–197, 2012.
  * [Lee18] John M. Lee.  Introduction to Riemannian manifolds, volume 176 of Graduate Texts in Mathematics.  Springer, Cham, second edition, 2018.
  * [Mat02] Michel Matthey.  Mapping the homology of a group to the $K$-theory of its $C^{*}$-algebra.  Illinois J. Math., 46(3):953–977, 2002.
  * [Mat12] Hiroki Matui.  Homology and topological full groups of étale groupoids on totally disconnected spaces.  Proc. Lond. Math. Soc. (3), 104(1):27–56, 2012.
  * [Mos68] G. D. Mostow.  Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms.  Inst. Hautes Études Sci. Publ. Math., (34):53–104, 1968.
  * [Phi00] N. Christopher Phillips.  A classification theorem for nuclear purely infinite simple $C^{*}$-algebras.  Doc. Math., 5:49–114, 2000.
  * [Ren08] Jean Renault.  Cartan subalgebras in $C^{*}$-algebras.  Irish Math. Soc. Bull., (61):29–63, 2008.
  * [RS87] Jonathan Rosenberg and Claude Schochet.  The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized $K$-functor.  Duke Math. J., 55(2):431–474, 1987.
  * [Tu99] Jean-Louis Tu.  La conjecture de Baum-Connes pour les feuilletages moyennables.  $K$-Theory, 17(3):215–264, 1999.


# Hilbert’s Theorem, via moving frames

William D. Dunbar

###### Abstract

We present a proof that the hyperbolic plane cannot be isometrically immersed
in Euclidean $3$-space by a $C^{\infty}$ map. Ideas from many topics in
(essentially) undergraduate mathematics are applied; the use of moving frames
and connection forms to express the geometry simplifies the outline of the
proof, compared to, say, using coordinate patches and Christoffel symbols. The
key transition is from expressions in terms of the principal directions on the
immersed surface (which give access to the Gaussian curvature) to expressions
in terms of the asymptotic directions (which yield a coordinate system and
give access to surface area).

##  1 Outline of the proof

  * •

Assume for simplicity that $\phi:\mathbb{H}^{2}\rightarrow\mathbb{R}^{3}$ is a
$C^{\infty}$ isometric embedding (immersions are no more difficult). In this
section, “the surface” will refer to the image of $\phi$.

  * •

Construct an orthonormal frame field $\\{e_{1},e_{2}\\}$ on $\mathbb{H}^{2}$
which is mapped by $\phi_{*}$ to unit vectors in the principal directions on
the surface (maximizing and minimizing normal curvature).

  * •

Construct a frame field $\\{E_{1},E_{2}\\}$ on $\mathbb{H}^{2}$ which is
mapped by $\phi_{*}$ to linearly independent unit vectors in asymptotic
directions (where the normal curvature equals zero).

  * •

Express the connection $1$-form $\omega_{1}^{2}$ (for $\\{e_{1},e_{2}\\}$)
first in terms of the $1$-forms dual to $\\{e_{1},e_{2}\\}$ and the principal
curvatures [Lemma 1], then in terms of the same forms and the angle $\alpha$
between either of the asymptotic directions and $e_{1}$ [Lemma 2], and finally
change basis to the $1$-forms dual to $\\{E_{1},E_{2}\\}$ [equation (23)].
(The Gaussian curvature on $\mathbb{H}^{2}$ being identically equal to $-1$ is
used in the proof of Lemma 2.)

  * •

Show that the dual $1$-forms $\\{\eta^{1},\eta^{2}\\}$ for $\\{E_{1},E_{2}\\}$
are closed forms, hence exact [Lemma 3]; furthermore, construct a (global!)
coordinate chart mapping $P\in\mathbb{H}^{2}$ to
$F(P)=(u^{1}(P),u^{2}(P))\in\\{(x^{1},x^{2})\\}=\mathbb{R}^{2}$, such that
$\frac{\partial}{\partial u^{1}}=E_{1}$ and $\frac{\partial}{\partial
u^{2}}=E_{2}$ [Proposition 3].

  * •

Define $\Theta$ to be the (oriented) angle between $E_{1}$ and $E_{2}$; hence,
$\sin\Theta$ is the area distortion factor when mapping from the
$x^{1}x^{2}$-plane to $\mathbb{H}^{2}$. Show that
$\sin\Theta=\frac{\partial^{2}\Theta}{\partial u^{1}\partial u^{2}}$ [equation
(26)].

  * •

Calculate the area in $\mathbb{H}^{2}$ corresponding to $[-a,a]\times[-a,a]$
in the $x^{1}x^{2}$-plane, and show that as $a\to\infty$, area is bounded
above by $2\pi$ [equation (2.4)] …but also calculate the area of
$\mathbb{H}^{2}$ by other means to show that it is infinite [equation (2.4)].
This is a contradiction, so $\phi$ cannot exist.

This approach to Hilbert’s theorem was inspired by the argument given by
Rubens Leão in the Appendix of [dC83]. Citations will be supplied for
equations that come from undergraduate differential geometry and/or from
tensor analysis, but proofs will usually be omitted. For differential
geometry, [O’N06] and [Shi18, Chapter 3, Section 3] use moving frames; [MP77]
and [dC16] may also be helpful. Basic facts about differential forms in
$\mathbb{R}^{n}$ (exterior differentiation, wedge product, pullback) can be
found in [dC94, Chapter 1] and in [Spi79a, Chapter 7].

##  2 Proof of the main theorem

######  Theorem 1.

There does not exist a $C^{\infty}$ isometric immersion of the hyperbolic
plane into Euclidean $3$-space.

Before starting the proof, we make a foundational definition and a remark on
notation.

######  Definition 1.

A _moving frame_ on an open subset $U$ of $\mathbb{R}^{n}$ is an ordered
$n$-tuple of smooth (i.e., $C^{\infty}$) vector fields
$\\{v_{1},\dots,v_{n}\\}$, such that the vectors at each point $p\in U$ form
an orthonormal basis for the tangent space $T_{p}(U)$.

The previous definition agrees with [dC83, page 118]; see also [dC94, page 77]
and [O’N06, Chapter II, Section 1]. Elsewhere, such as in [Spi79b, page 285],
the vectors in a moving frame need only be linearly independent (and span).

######  Remark.

We will use lower subscripts for contravariant objects such as vectors, and
upper subscripts for their components with respect to a basis (as in the
preceding definition). Similarly, we will use upper subscripts for covariant
objects such as differential forms, and lower subscripts for their components
with respect to a basis. Summation signs are needed only a few times, and they
are _not_ omitted, as they would be in “Einstein summation convention”
[Spi79a, pages 50–51, pages 155–158].

\begin{overpic}[scale={0.36}]{AllMapsR9a.pdf} \put(12.0,4.0){$\mathbb{R}^{2}$}
\put(62.0,4.0){$\mathbb{D}$} \put(88.0,4.0){$\mathbb{R}^{3}$}
\put(26.5,18.0){\LARGE$\overset{F}{\longleftarrow}$}
\put(69.5,18.0){\LARGE$\overset{\phi}{\longrightarrow}$}
\put(19.0,20.5){$x^{1}$} \put(13.5,25.5){$x^{2}$} \put(46.0,14.0){$P$}
\put(54.5,19.8){$e_{1}$} \put(44.5,22.5){$e_{2}$} \put(54.4,15.8){$E_{1}$}
\put(50.3,22.3){$E_{2}$} \end{overpic} Figure 1: Overview

###### Proof of Theorem 1.

The Poincaré disk $\mathbb{D}$ will represent the hyperbolic plane; define
$O:=(0,0)\in\mathbb{D}$ (see the Appendix for the Riemannian metric and other
details). Suppose that $\phi:\mathbb{D}\longrightarrow\mathbb{R}^{3}$ is a
$C^{\infty}$ isometric immersion.

###  2.1 Construct vector fields and $1$-forms on $\mathbb{D}$

For all $P\in\mathbb{D}$, there is a positive number $\epsilon_{P}$, such that
$\phi$ is an embedding, when restricted to the open disk $U_{P}$ of radius
$\epsilon_{P}$ in the hyperbolic metric. The tangent space to the surface
$\phi(U_{P})=:\bar{U}_{P}$ at $\phi(P)=:\bar{P}$ can inherit an orientation
from $T_{P}(\mathbb{D})$, via $(\phi_{*})_{P}$, so there is a preferred unit
normal vector at $\bar{P}$, which can be continuously (indeed, smoothly)
extended to a unit normal vector field $\bar{e}_{3}:\bar{U}_{P}\rightarrow
S^{2}$.

At every point $Q\in U_{P}$, the map
$(-d\bar{e}_{3})_{\bar{Q}}:T_{\bar{Q}}(\bar{U}_{P})\rightarrow
T_{\bar{e}_{3}(\bar{Q})}S^{2}$, is a self-adjoint linear transformation of
$T_{\bar{Q}}(\bar{U}_{P})=\phi_{*}(T_{Q}(\mathbb{D}))$ [dC16, page 142]; note
that $T_{\bar{e}_{3}(\bar{Q})}S^{2}$ and $T_{\bar{Q}}(\bar{U}_{P})$ are both
names for the orthogonal complement in $\mathbb{R}^{3}$ of
$\operatorname{span}(\bar{e}_{3}(\bar{Q}))$. The eigenvalues are equal to the
principal curvatures (let $\bar{\kappa}_{1}(\bar{Q})$ denote the positive
eigenvalue, and $\bar{\kappa}_{2}(\bar{Q})$ the negative eigenvalue), and the
corresponding (one-dimensional, orthogonal) eigenspaces
$\bar{L}_{1}(\bar{Q}),\bar{L}_{2}(\bar{Q})$ give the principal directions
[dC16, page 146]. At this stage, we are using the fact that the Gaussian
curvature (product of the principal curvatures) is negative at every point of
$\bar{U}_{P}$, but _not_ (yet) the fact that it is constant and equal to $-1$.

Via $(\phi|_{U_{P}})_{*}^{-1}$, we can transfer these eigenspaces back to form
two smooth line fields on each $U_{P}$. Whenever $R\in U_{P}\cap U_{Q}$, the
line fields for $U_{P}$ and $U_{Q}$ will agree at $R$, since they both
describe the extrinsic geometry of $\phi(U_{P}\cap U_{Q})$ at $\phi(R)$. Hence
we can merge all of the “local” line fields into two “global” line fields
$L_{1},L_{2}$ on $\mathbb{D}$. There is a two-sheeted covering space of
$\mathbb{D}$, consisting of all the unit vectors in $L_{1}$, and since
$\mathbb{D}$ is simply-connected, this covering space is not connected. In
other words, there are two unit vector fields $\pm e_{1}$ on $\mathbb{D}$
which everywhere point along $L_{1}$. We define $e_{2}$ to be the result of
rotating $e_{1}$ a quarter-turn counter-clockwise (i.e., $+\pi/2$ radians).

This allows us to define $\bar{e}_{1}:=\phi_{*}(e_{1})$ and
$\bar{e}_{2}:=\phi_{*}(e_{2})$ on each $\bar{U}_{P}$, so we now have a
(positively oriented) moving frame $\\{\bar{e}_{1},\bar{e}_{2},\bar{e}_{3}\\}$
on $\bar{U}_{P}$. As in [dC94, page 82], we can extend this moving frame to a
small open subset $W_{P}$ of $\mathbb{R}^{3}$ by displacing frames normally by
a sufficiently small amount (choose $\nu_{P}>0$ such that
$U_{P}\times(-\nu_{P},\nu_{P})\rightarrow W_{P}$ by
$(Q,t)\mapsto\bar{Q}+t\cdot[\bar{e}_{3}]_{\bar{Q}}$ is a diffeomorphism). See
Figure 2 for an illustration.

Since $d\bar{e}_{1},d\bar{e}_{2},d\bar{e}_{3}$ are linear transformations on
$\mathbb{R}^{3}$ at every point of $W_{P}$ (in other words, they are
$\mathbb{R}^{3}$-valued $1$-forms, as defined in [Spi79a, page 546], while
$\bar{e}_{1},\bar{e}_{2},\bar{e}_{3}$ are $\mathbb{R}^{3}$-valued $0$-forms),
we can define the _connection $1$-forms_ $\bar{\omega}^{k}_{j}\ (1\leq j,k\leq
3)$ by

$d\bar{e}_{j}(v)=\sum_{k=1}^{3}\bar{\omega}^{k}_{j}(v)\ \bar{e}_{k},$ (1)

for all vector fields $v$ on $W_{P}$ [dC94, page 78] [Spi79b, page 287]. The
notation $\bar{\omega}_{j}^{k}$ follows [Spi79b, page 287] and corresponds to
$\omega_{jk}$ in [dC94, page 78].

On each $W_{P}$ we can also define $1$-forms
$\bar{\theta}^{1},\bar{\theta}^{2},\bar{\theta}^{3}$ dual to the frame, that
is,

$\bar{\theta}^{i}(\bar{e}_{j})=\delta^{i}_{j};\quad 1\leq i,j\leq 3$ (2)

(using the Kronecker delta). These $1$-forms on $W_{P}$ are related to each
other by the Cartan structure equations, which follow.

######  Proposition 1.

For all $1\leq i,j\leq 3$, and for any moving frame on an open subset of
$\mathbb{R}^{3}$, with connection forms and dual forms defined as in equations
(1) and (2),

$\displaystyle d\bar{\theta}^{i}$
$\displaystyle=-\sum_{k=1}^{3}\bar{\omega}^{i}_{k}\wedge\bar{\theta}^{k}$ (3)
$\displaystyle d\bar{\omega}^{i}_{j}$
$\displaystyle=-\sum_{k=1}^{3}\bar{\omega}^{i}_{k}\wedge\bar{\omega}^{k}_{j}$
(4)

###### Proof.

See [Spi79b, page 287] or [dC94, page 78]. The former obtains (3) by expanding
$0=d^{2}I$ as a sum of scalar multiples of the $\\{\bar{e}_{i}\\}$, where
$I:\mathbb{R}^{3}\longrightarrow\mathbb{R}^{3}$ is the identity map (and
$dI=\sum_{i=1}^{3}\bar{\theta}^{i}\wedge\bar{e}_{i}$, here an equation of
$\mathbb{R}^{3}$-valued $1$-forms). Similarly, (4) comes from
$0=d^{2}\bar{e}_{j}$. ∎

\begin{overpic}[height=126.47249pt]{CloseupU1bT.pdf}
\put(31.0,15.0){$\bar{e}_{1}$} \put(68.0,31.0){$\bar{e}_{2}$}
\put(45.0,47.0){$\bar{e}_{3}$} \put(53.0,34.0){$\bar{P}$} \end{overpic}

Figure 2: At left, $\bar{U}_{P}=\phi(U_{P})$ and frame at $\bar{P}=\phi(P)$;
at right, $\bar{U}_{P}\subset W_{P}$.

Now, on each $U_{P}$, we can define the $1$-forms
$\theta^{i}:=\phi^{*}\bar{\theta}^{i}$, $1\leq i\leq 3$ (noting that
$\bar{\theta}^{3}$ pulls back to the zero $1$-form) and
$\omega^{i}_{j}:=\phi^{*}\bar{\omega}^{i}_{j}$, $1\leq i,j\leq 3$. The reader
can easily check that $\theta^{1},\theta^{2}$ are dual to $e_{1},e_{2}$.

######  Remark.

Orthonormality of the moving frame $\\{\bar{e}_{1},\bar{e}_{2},\bar{e}_{3}\\}$
is, in fact, not required for equations (3) and (4); linear independence of
the frame field suffices. However, orthonormality implies the relations
$\bar{\omega}^{j}_{i}=-\bar{\omega}^{i}_{j}$, hence
$\omega^{j}_{i}=-\omega^{i}_{j}$, and in particular, $\omega^{i}_{i}=0$
[Spi79b, page 292] [dC94, page 78].

Since $\\{e_{1},e_{2}\\}$ are globally defined on $\mathbb{D}$, the moving
frames $\\{\bar{e}_{1},\bar{e}_{2},\bar{e}_{3}\\}$ at
$\bar{R}\in\bar{U}_{P}\cap\bar{U}_{Q}$, arising respectively from
$\phi|_{U_{P}}$ and $\phi|_{U_{Q}}$, will agree. Hence the $1$-forms
$\theta^{i}$ and $\omega^{i}_{j}$, defined above, will agree on $U_{P}\cap
U_{Q}$. Consequently, as we did above with the line fields, we merge the
“local” $\theta$’s and $\omega$’s into “global” smooth $1$-forms that are
defined on $\mathbb{D}$.

Furthermore, since the “unbarred” versions of equations (3) and (4) can be
justified by basic facts about pullbacks and wedge products in each $U_{P}$,
Proposition 1 implies the following equations on $\mathbb{D}$ (or more
precisely, in the exterior algebra of $C^{\infty}$ differential forms on
$\mathbb{D}$):

$\begin{cases}d\theta^{1}&=-(\omega^{1}_{1}\wedge\theta^{1}+\omega^{1}_{2}\wedge\theta^{2}+\omega^{1}_{3}\wedge\theta^{3})=-0\wedge\theta^{1}-\omega^{1}_{2}\wedge\theta^{2}-\omega^{1}_{3}\wedge
0=\omega_{1}^{2}\wedge\theta^{2}\\\
d\theta^{2}&=-(\omega^{2}_{1}\wedge\theta^{1}+\omega^{2}_{2}\wedge\theta^{2}+\omega^{2}_{3}\wedge\theta^{3})=\dots=-\omega_{1}^{2}\wedge\theta^{1}\end{cases}$
(5)

$\begin{cases}d\omega_{1}^{2}&=-(\omega_{1}^{2}\wedge\omega_{1}^{1}+\omega_{2}^{2}\wedge\omega_{1}^{2}+\omega_{3}^{2}\wedge\omega_{1}^{3})=\omega_{2}^{3}\wedge\omega_{1}^{3}=-\omega_{1}^{3}\wedge\omega_{2}^{3}\\\
d\omega_{1}^{3}&=-(\omega_{1}^{3}\wedge\omega_{1}^{1}+\omega_{2}^{3}\wedge\omega_{1}^{2}+\omega_{3}^{3}\wedge\omega_{1}^{3})=-\omega_{2}^{3}\wedge\omega_{1}^{2}=\omega_{1}^{2}\wedge\omega_{2}^{3}\\\
d\omega_{2}^{3}&=-(\omega_{1}^{3}\wedge\omega_{2}^{1}+\omega_{2}^{3}\wedge\omega_{2}^{2}+\omega_{3}^{3}\wedge\omega_{2}^{3})=\omega_{1}^{3}\wedge\omega_{1}^{2}=-\omega_{1}^{2}\wedge\omega_{1}^{3}\end{cases}$
(6)

Also, by definition, in each $\bar{U}_{P}$,

$d\bar{e}_{3}(\bar{e}_{1})=-\bar{\kappa}_{1}\bar{e}_{1}\textrm{ and
}d\bar{e}_{3}(\bar{e}_{2})=-\bar{\kappa}_{2}\bar{e}_{2}$ (7)

[BL10, page 179], [MP77, pages 125–129], [Spi79b, pages 103–106]. From the
first equation we have, using (1),
$\bar{\omega}^{1}_{3}(\bar{e}_{1})=-\bar{\kappa}_{1}$ and
$\bar{\omega}^{2}_{3}(\bar{e}_{1})=0$; similarly, the second equation implies
$\bar{\omega}^{1}_{3}(\bar{e}_{2})=0$ and
$\bar{\omega}^{2}_{3}(\bar{e}_{2})=-\bar{\kappa}_{2}$. We conclude, after
pullback to $\mathbb{D}$, with $\kappa_{i}:=\bar{\kappa}_{i}\circ\phi\
(i=1,2)$, that

$\omega_{3}^{1}=-\kappa_{1}\theta^{1}\textrm{ and
}\omega_{3}^{2}=-\kappa_{2}\theta^{2},\textrm{ so
}\omega_{1}^{3}=\kappa_{1}\theta^{1}\textrm{ and
}\omega_{2}^{3}=\kappa_{2}\theta^{2}.$ (8)

######  Remark.

To keep the signs straight, it is useful to keep in mind the example of the
unit sphere (with one point removed, so that there exists a moving frame),
oriented with outward normal. Then $\bar{e}_{3}$ equals $p$ at $p\in S^{2}$,
so $d\bar{e}_{3}$ is the identity map. The normal curvatures at each point all
equal $-1$ (negative since curves on the sphere bend _away_ from the normal
vector). So for this surface the principal curvatures both equal $-1$, which
is consistent with equation (7).

###  2.2 Expressions for $\omega_{1}^{2}$ in terms of
$\\{\theta^{1},\theta^{2}\\}$

We now turn to the proofs of two key lemmas, which relate $\omega_{1}^{2}$ to
the principal curvatures and to the angle between the asymptotic directions
and $e_{1}$.

######  Lemma 1.

With $\\{e_{1},e_{2}\\}$, $\\{\theta^{1},\theta^{2}\\}$, and
$\omega_{1}^{2},\kappa_{1},\kappa_{2}$ defined as above,

$(\kappa_{1}-\kappa_{2})\omega_{1}^{2}=d\kappa_{1}(e_{2})\theta^{1}+d\kappa_{2}(e_{1})\theta^{2}.$
(9)

###### Proof.

To verify this equation of $1$-forms, it suffices to show the following two
equations:

$(\kappa_{1}-\kappa_{2})\omega_{1}^{2}(e_{1})\overset{?}{=}d\kappa_{1}(e_{2})\textrm{
and
}(\kappa_{1}-\kappa_{2})\omega_{1}^{2}(e_{2})\overset{?}{=}d\kappa_{2}(e_{1})$
(10)

We rewrite $d\omega_{1}^{3}$ in two ways, using equations (5), (6), and (8),
obtaining

$\kappa_{1}\omega_{1}^{2}\wedge\theta^{2}+d\kappa_{1}\wedge\theta^{1}\overset{\eqref{Equ:ddual}}{=}\kappa_{1}d\theta^{1}+d\kappa_{1}\wedge\theta^{1}\overset{\eqref{Equ:princurv}}{=}d\omega_{1}^{3}\overset{\eqref{Equ:dconn}}{=}\omega_{1}^{2}\wedge\omega_{2}^{3}\overset{\eqref{Equ:princurv}}{=}\kappa_{2}\omega_{1}^{2}\wedge\theta^{2}.$

In other words,
$(\kappa_{1}-\kappa_{2})\omega_{1}^{2}\wedge\theta^{2}=-d\kappa_{1}\wedge\theta^{1}$.
Evaluating both sides of this equation on the pair ($e_{1}$,$e_{2}$), we
obtain

$\displaystyle(\kappa_{1}-\kappa_{2})\omega_{1}^{2}\wedge\theta^{2}(e_{1},e_{2})$
$\displaystyle=(\kappa_{1}-\kappa_{2})(\omega_{1}^{2}(e_{1})\
\theta^{2}(e_{2})-\omega_{1}^{2}(e_{2})\
\theta^{2}(e_{1}))=(\kappa_{1}-\kappa_{2})\omega_{1}^{2}(e_{1})$
$\displaystyle\textrm{ and }-d\kappa_{1}\wedge\theta^{1}(e_{1},e_{2})$
$\displaystyle=-d\kappa_{1}(e_{1})\ \theta^{1}(e_{2})+d\kappa_{1}(e_{2})\
\theta^{1}(e_{1})=d\kappa_{1}(e_{2}),$

which justifies the first half of (10). Similarly, rewriting
$d\omega_{2}^{3}$,

$-\kappa_{2}\omega_{1}^{2}\wedge\theta^{1}+d\kappa_{2}\wedge\theta^{2}\overset{\eqref{Equ:ddual}}{=}\kappa_{2}d\theta^{2}+d\kappa_{2}\wedge\theta^{2}\overset{\eqref{Equ:princurv}}{=}d\omega_{2}^{3}\overset{\eqref{Equ:dconn}}{=}-\omega_{1}^{2}\wedge\omega_{1}^{3}\overset{\eqref{Equ:princurv}}{=}-\kappa_{1}\omega_{1}^{2}\wedge\theta^{1}.$

In other words,
$-(\kappa_{1}-\kappa_{2})\omega_{1}^{2}\wedge\theta^{1}=d\kappa_{2}\wedge\theta^{2}$.
Again evaluating the left-hand and right-hand sides at ($e_{1}$,$e_{2}$),

$\displaystyle-(\kappa_{1}-\kappa_{2})\omega_{1}^{2}\wedge\theta^{1}(e_{1},e_{2})$
$\displaystyle=\dots=(\kappa_{1}-\kappa_{2})\omega_{1}^{2}(e_{2})$
$\displaystyle\textrm{ and }d\kappa_{2}\wedge\theta^{2}(e_{1},e_{2})$
$\displaystyle=\dots=d\kappa_{2}(e_{1}),$

which justifies the second half of (10). ∎

For all $P\in\mathbb{D}$, the normal curvatures at $\phi(P)$ can be described
by
$\operatorname{II}(v):=\langle-d\bar{e}_{3}(\phi_{*}(v)),\phi_{*}(v)\rangle$,
defined in terms of the Euclidean inner product for vectors in
$\mathbb{R}^{3}$ [dC94, page 87]. In essence, this is the second fundamental
form for the immersed surface near $\phi(P)$, pulled back to a quadratic form
on $T_{P}(\mathbb{D})$. Using Euler’s theorem [dC16, page 147] [MP77, page
129], and (for the first time) making use of the fact that Gaussian curvature
$K:=\kappa_{1}\kappa_{2}$ equals $-1$ everywhere on $\mathbb{D}$, we have
normal curvature zero when

$0=\operatorname{II}(v)=\kappa_{1}\cos^{2}\alpha+\kappa_{2}\sin^{2}\alpha=\kappa_{1}\cos^{2}\alpha-(1/\kappa_{1})\sin^{2}\alpha$

for a unit vector $v$ that makes an angle $\pm\alpha$ with $e_{1}$. We
conclude that $\tan^{2}\alpha=\kappa_{1}^{2}$, so set
$\alpha:=\arctan(\kappa_{1})\in(0,\pi/2)$, and define the asymptotic
directions

$E_{1}:=\cos\alpha\cdot e_{1}-\sin\alpha\cdot e_{2}\textrm{ and
}E_{2}:=\cos\alpha\cdot e_{1}+\sin\alpha\cdot e_{2}.$ (11)

It follows that

$e_{1}=\frac{1}{2}\sec\alpha(E_{1}+E_{2})\textrm{ and
}e_{2}=\frac{1}{2}\csc\alpha(-E_{1}+E_{2}).$ (12)

######  Lemma 2.

With $\\{e_{1},e_{2}\\}$, $\\{\theta^{1},\theta^{2}\\}$, and
$\omega_{1}^{2},\alpha$ defined as above,

$\omega_{1}^{2}=\tan\alpha\,d\alpha(e_{2})\cdot\theta^{1}+\cot\alpha\,d\alpha(e_{1})\cdot\theta^{2}$
(13)

###### Proof.

Since $\kappa_{2}=-1/\kappa_{1}=-1/\tan\alpha=-\cot\alpha$,

$\kappa_{1}-\kappa_{2}=\frac{\sin\alpha}{\cos\alpha}+\frac{\cos\alpha}{\sin\alpha}=\frac{1}{\cos\alpha\,\sin\alpha}.$

Therefore, using $d\kappa_{1}=d(\tan\alpha)=\sec^{2}\alpha\,d\alpha$ and
$d\kappa_{2}=d(-\cot\alpha)=\csc^{2}\alpha\,d\alpha$,

$\displaystyle\frac{d\kappa_{1}}{\kappa_{1}-\kappa_{2}}$
$\displaystyle=(\sec^{2}\alpha\,d\alpha)(\cos\alpha\,\sin\alpha)=\tan\alpha\,d\alpha$
(14) $\displaystyle\frac{d\kappa_{2}}{\kappa_{1}-\kappa_{2}}$
$\displaystyle=(\csc^{2}\alpha\,d\alpha)(\cos\alpha\,\sin\alpha)=\cot\alpha\,d\alpha$
(15)

After evaluating (14) at $e_{2}$ and (15) at $e_{1}$, the desired conclusion
follows from Lemma 1. ∎

###  2.3 Use $\\{E_{1},E_{2}\\}$ to construct a global coordinate map

We now turn our attention to the properties of the asymptotic vector fields
$E_{1},E_{2}$.

######  Lemma 3.

Let $\\{\eta^{1},\eta^{2}\\}$ be the $1$-forms which are dual to
$\\{E_{1},E_{2}\\}$. Then $d\eta^{1}=0$ and $d\eta^{2}=0$.

###### Proof.

It is not hard to use the definitions of $E_{1}$ and $E_{2}$ in (11) to verify
that

$\eta^{1}=\frac{1}{2}[\sec\alpha\cdot\theta^{1}-\csc\alpha\cdot\theta^{2}]\textrm{
and
}\eta^{2}=\frac{1}{2}[\sec\alpha\cdot\theta^{1}+\csc\alpha\cdot\theta^{2}]$
(16)

Next, we see that

$\displaystyle d(\sec\alpha\cdot\theta^{1})$
$\displaystyle\overset{\eqref{Equ:ddual}}{=}\sec\alpha\,\tan\alpha\,d\alpha\wedge\theta^{1}+\sec\alpha(\omega_{1}^{2}\wedge\theta^{2})$
$\displaystyle=\sec\alpha\,\tan\alpha\,(d\alpha(e_{1})\cdot\theta^{1}+d\alpha(e_{2})\cdot\theta^{2})\wedge\theta^{1}+\sec\alpha(\omega_{1}^{2}\wedge\theta^{2})$
$\displaystyle=-\sec\alpha\,\tan\alpha\,d\alpha(e_{2})\cdot\theta^{1}\wedge\theta^{2}+\sec\alpha(\omega_{1}^{2}\wedge\theta^{2})$
$\displaystyle\overset{\eqref{Equ:conntancot}}{=}-\sec\alpha\,\tan\alpha\,d\alpha(e_{2})\cdot\theta^{1}\wedge\theta^{2}$
$\displaystyle\qquad+\sec\alpha((\tan\alpha\,d\alpha(e_{2})\cdot\theta^{1}+\cot\alpha\,d\alpha(e_{1})\cdot\theta^{2})\wedge\theta^{2})$
$\displaystyle=-\sec\alpha\,\tan\alpha\,d\alpha(e_{2})\cdot\theta^{1}\wedge\theta^{2}+\sec\alpha(\tan\alpha\,d\alpha(e_{2})\cdot\theta^{1}\wedge\theta^{2})=0$
(17)

and similarly

$\displaystyle d(\csc\alpha\cdot\theta^{2})$
$\displaystyle\overset{\eqref{Equ:ddual}}{=}-\csc\alpha\,\cot\alpha\,d\alpha\wedge\theta^{2}+\csc\alpha(-\omega_{1}^{2}\wedge\theta^{1})$
$\displaystyle=-\csc\alpha\,\cot\alpha\,(d\alpha(e_{1})\cdot\theta^{1}+d\alpha(e_{2})\cdot\theta^{2})\wedge\theta^{2}+\csc\alpha(-\omega_{1}^{2}\wedge\theta^{1})$
$\displaystyle=-\csc\alpha\,\cot\alpha\,d\alpha(e_{1})\cdot\theta^{1}\wedge\theta^{2}-\csc\alpha(\omega_{1}^{2}\wedge\theta^{1})$
$\displaystyle\overset{\eqref{Equ:conntancot}}{=}-\csc\alpha\,\cot\alpha\,d\alpha(e_{1})\cdot\theta^{1}\wedge\theta^{2}$
$\displaystyle\qquad-\csc\alpha((\tan\alpha\,d\alpha(e_{2})\cdot\theta^{1}+\cot\alpha\,d\alpha(e_{1})\cdot\theta^{2})\wedge\theta^{1})$
$\displaystyle=-\csc\alpha\,\cot\alpha\,d\alpha(e_{1})\cdot\theta^{1}\wedge\theta^{2}-\csc\alpha(\cot\alpha\,d\alpha(e_{1})\cdot(-\theta^{1}\wedge\theta^{2}))=0$
(18)

By subtracting and adding equations (2.3) and (2.3), and then comparing to
equation (16), we conclude that $d\eta^{1}=0$ and $d\eta^{2}=0$. ∎

We next show that the vector fields $E_{1}$ and $E_{2}$ on $\mathbb{D}$ each
generate a global flow (i.e., each integral curve is defined for all
$t\in\mathbb{R}$). We will use the following special case of [Cod61, Chapter
6, §6, Theorem 2].

######  Theorem 2.

Let $a,b$ be positive numbers, let $t_{0}\in\mathbb{R}$, let
$\mathbf{y_{0}}\in\mathbb{R}^{2}$, and let $R:=\\{(t,\mathbf{y}):|t-t_{0}|\leq
a,|\mathbf{y}-\mathbf{y_{0}}|\leq b\\}$. If

  1. 1.

$f:R\rightarrow\mathbb{R}^{2}$ is a continuous function, and

  2. 2.

there exists $L>0$ such that for all $(t,\mathbf{y_{1}}),(t,\mathbf{y_{2}})\in
R$, $|f(t,\mathbf{y_{1}})-f(t,\mathbf{y_{2}})|\leq
L|\mathbf{y_{1}}-\mathbf{y_{2}}|$, and

  3. 3.

there exists $M>0$ such that for all $(t,\mathbf{y})\in R$,
$|f(t,\mathbf{y})|\leq M$,

then the initial-value problem $d\mathbf{y}/dt=f(t,\mathbf{y}),\
\mathbf{y}(t_{0})=\mathbf{y_{0}}$ has a solution

$\mathbf{y}=\gamma(t),\
\gamma:(t_{0}-\epsilon,t_{0}+\epsilon)\rightarrow\\{\mathbf{y}\in\mathbb{R}^{2}:|\mathbf{y}-\mathbf{y_{0}}|\leq
b\\},$

where $\epsilon:=\min(a,b/M)$. ∎

Note that $\epsilon$ does not depend on $L$, the Lipschitz constant.

######  Proposition 2.

If $E$ is a $C^{\infty}$ vector field on $\mathbb{D}$ which everywhere has
unit length with respect to the hyperbolic metric, then every maximal integral
curve of $E$ is defined for all $t\in\mathbb{R}$.

###### Proof.

Preparing to apply Theorem 2, we will think of the unit-hyperbolic-length
vector field $E$ as a vector field on the open unit disk in $\mathbb{R}^{2}$,
and now measure length with respect to the Euclidean metric. Set $t_{0}:=0$
and $\mathbf{y_{0}}:=(0,0)=O$. Next, set $a:=1$, and set $b:=1/2$, so that
$R=\\{|t|\leq 1,|\mathbf{y}|\leq 1/2\\}$, and define
$f(t,\mathbf{y}):=E|_{\mathbf{y}}$. By the definition of the hyperbolic metric
on $\mathbb{D}$ (see Appendix), we can take $M=1/2$ (for any choice of
$b\in(0,1)$). Hence, $\epsilon=\min(1,\frac{1}{2}/\frac{1}{2})=1$.

The partial derivatives of $f$ with respect to the standard coordinates of
$\mathbb{R}^{2}$ are continuous on the compact set $R$, and hence are bounded,
so we conclude that the Lipschitz constant $L$ exists. By Theorem 2, we have a
solution $\gamma:(-1,1)\rightarrow\mathbb{D}\subset\mathbb{R}^{2}$.

Now fix an arbitrary point $P\in\mathbb{D}$. There exists a hyperbolic
isometry $\psi:\mathbb{D}\rightarrow\mathbb{D}$ which maps $P$ to $O$ (see
equation (30)). Use $\psi_{*}$ to map the vector field $E$ to a “recentered”
vector field $\tilde{E}$ on $\mathbb{D}$, then apply the previous reasoning to
$\tilde{E}$, obtaining an integral curve $\gamma:(-1,1)\rightarrow\mathbb{D}$
such that $\gamma(0)=O$. The curve
$\psi^{-1}\circ\gamma:(-1,1)\rightarrow\mathbb{D}$ will be the desired
integral curve through $P$.

Since $E$ is a smooth vector field on the smooth manifold $\mathbb{D}$, and
every point of $\mathbb{D}$ can flow forward and backward along $E$ for at
least one unit of time, it follows from the Uniform Time Lemma [Lee13, Lemma
9.15, page 216] that every integral curve can be extended to an integral curve
defined on $(-\infty,\infty)$. ∎

######  Corollary 1.

All integral curves of $E_{1}$ and $E_{2}$, the unit asymptotic directions,
are defined for all time. ∎

######  Proposition 3.

There is a $C^{\infty}$ diffeomorphism $F$ from $\mathbb{D}$ onto
$\mathbb{R}^{2}:=\\{(x^{1},x^{2}):x^{1},x^{2}\in\mathbb{R}\\}$, such that

  1. 1.

$F$ is a (global) coordinate chart for $\mathbb{D}$, and

  2. 2.

$F_{*}(E_{i})=\frac{\partial}{\partial x^{i}}$, $i=1,2$.

###### Proof.

Working in $\mathbb{D}$, with $O=(0,0)$, define
$F(P):=(\int_{O}^{P}\eta^{1},\int_{O}^{P}\eta^{2})$. $F$ is well-defined,
using Green’s theorem, since $\eta^{1}$ and $\eta^{2}$ are closed and any two
paths from $O$ to $P$ bound a region in $\mathbb{D}$. By definition, the
differential of $F$ at $P$ will map $(E_{i})_{P}$ to $(\partial/\partial
x^{i})_{F(P)}$, for $i=1,2$. Since $E_{1}$ and $E_{2}$ are everywhere linearly
independent, the differential is everywhere nonsingular, so $F$ is a local
$C^{\infty}$ diffeomorphism (using a $C^{\infty}$ version of the Inverse
Function theorem; see [Lee13, Theorem C.34] ).

In addition, $F$ is surjective, since in order to find a point in $\mathbb{D}$
which maps to $(a,b)\in\mathbb{R}^{2}$, it suffices to start at $O$ at time
zero, follow an integral curve for $E_{1}$ (forward if $a>0$, backward if
$a<0$) for time $|a|$, then similarly follow an integral curve for $E_{2}$ for
time $|b|$, using Corollary 1. By the definition of $F$, along the first part
of the curve, the second coordinate of the output will not change, and vice
versa.

With respect to the orthonormal bases $\\{e_{1},e_{2}\\}$ in $\mathbb{D}$ and
$\\{\partial/\partial x^{1},\partial/\partial x^{2}\\}$ in $\mathbb{R}^{2}$,
any local inverse for $F$ has Jacobian matrix

$M_{\alpha}:=\begin{bmatrix}\cos\alpha&\cos\alpha\\\
-\sin\alpha&\sin\alpha\end{bmatrix},$ (19)

(using equation (11)) and the singular values of this matrix are easily
calculated to be $\sqrt{2}\cos\alpha$ and $\sqrt{2}\sin\alpha$. From this we
can conclude that all local inverses are Lipschitz maps with constant
$\sqrt{2}$. Now let $\gamma:[0,1]\rightarrow\mathbb{R}^{2}$ be a continuous
path, and let $Q$ be a point in $\mathbb{D}$ such that $F(Q)=\gamma(0)$. We
claim that $\gamma$ has a unique lift to $Q$; that is, there is a unique path
$\tilde{\gamma}:[0,1]\rightarrow\mathbb{D}$ such that $\tilde{\gamma}(0)=Q$
and such that $F\circ\tilde{\gamma}=\gamma$.

Let $\mathcal{T}:=\\{t\in[0,1]:\gamma|_{[0,t]}\textrm{ has a unique lift}\\}$.
This is an open subset of $[0,1]$, since when $\gamma$ lifts as far as
$[0,s]$, a local diffeomorphism from an open subset of $\mathbb{D}$ to an open
neighborhood of $\gamma(s)$ will allow the lift to be (uniquely) extended a
little further. It is also true that $\mathcal{T}$ is a closed subset of
$[0,1]$, since if $\gamma|_{[0,s)}$ lifts, then $\gamma$ is uniformly
continuous on $[0,s)\subset[0,1]$. Using the Lipschitz bound in the previous
paragraph, $\tilde{\gamma}|_{[0,s)}$ is also uniformly continuous, and hence
(see Appendix for details) $\tilde{\gamma}$ has a unique extension to $[0,s]$.
Finally, since $[0,1]$ is connected, and $\mathcal{T}$ is a non-empty subset
that is both open and closed, we conclude that $\mathcal{T}=[0,1]$, hence
$\gamma$ lifts uniquely.

Since $\mathbb{D}$ is path-connected, and $\mathbb{R}^{2}$ is simply-
connected, the following lemma applies to the map
$F:\mathbb{D}\rightarrow\mathbb{R}^{2}$.

######  Lemma 4 ([Lim03, Proposition 6.12]).

Let $f:X\rightarrow Y$ be a local homeomorphism with the unique-path-lifting
property. If $X$ is path-connected and $Y$ is simply-connected, then $f$ is a
homeomorphism.

###### Proof.

(Sketch) Construct a path between any two points in $X$ that $f$ maps to the
same place. Use $f$ to map the path, continuously deform the mapped path to a
constant path, and then lift the homotopy, to show that the initial and final
points of the original path are equal. ∎

Since $F$ is now known to be a homeomorphism (and in particular is injective),
and has already been shown to be a local $C^{\infty}$ diffeomorphism, it
follows that $F$ is a $C^{\infty}$ diffeomorphism. ∎

######  Corollary 2.

Let $G:\mathbb{R}^{2}\rightarrow\mathbb{D}$ be defined by mapping the point
$(x^{1},x^{2})\in\mathbb{R}^{2}$ to the point in $\mathbb{D}$ obtained by
starting at $O$, following an integral path for the vector field $E_{1}$ for
time $x^{1}$, then following an integral path for the vector field $E_{2}$ for
time $x^{2}$ (moving backward when these numbers are negative). Then
$G=F^{-1}$.

###### Proof.

By construction, $F\circ G$ is the identity map on $\mathbb{R}^{2}$.
Proposition 3 implies that $F$ is injective, so $F\circ
I_{\mathbb{D}}=I_{\mathbb{R}^{2}}\circ F=(F\circ G)\circ F=F\circ(G\circ F)$
implies $I_{\mathbb{D}}=G\circ F$. ∎

Since Proposition 3 has shown that $F$ is a coordinate mapping for
$\mathbb{D}$, we can define coordinate functions $u^{1},u^{2}$ on $\mathbb{D}$
such that $u^{1}=x^{1}\circ F,\ u^{2}=x^{2}\circ F$ and

$\frac{\partial}{\partial u^{1}}=E_{1},\ \frac{\partial}{\partial
u^{2}}=E_{2};\textrm{ hence }du^{1}=\eta^{1},\ du^{2}=\eta^{2}.$ (20)

(By a common abuse of notation, we let $x^{1}$ and $x^{2}$ denote the
functions on $\mathbb{R}^{2}$ that return, respectively, the first and second
coordinates of a point.)

###  2.4 Area computations

Adding and subtracting the equations in (16), we have

$\theta^{1}=\cos\alpha(\eta^{1}+\eta^{2})\textrm{ and
}\theta^{2}=\sin\alpha(-\eta^{1}+\eta^{2}).$ (21)

Therefore, using Lemma 2,

$\displaystyle\omega_{1}^{2}$
$\displaystyle=\tan\alpha\,d\alpha(e_{2})\cdot\theta^{1}+\cot\alpha\,d\alpha(e_{1})\cdot\theta^{2}$
$\displaystyle\overset{\eqref{Equ:thetafrometa}}{=}\tan\alpha\,d\alpha(e_{2})\cdot\cos\alpha(\eta^{1}+\eta^{2})+\cot\alpha\,d\alpha(e_{1})\cdot\sin\alpha(-\eta^{1}+\eta^{2})$
$\displaystyle=\sin\alpha\,d\alpha(e_{2})\cdot(\eta^{1}+\eta^{2})+\cos\alpha\,d\alpha(e_{1})\cdot(-\eta^{1}+\eta^{2})$
$\displaystyle=(\sin\alpha\,d\alpha(e_{2})-\cos\alpha\,d\alpha(e_{1}))\eta^{1}+(\sin\alpha\,d\alpha(e_{2})+\cos\alpha\,d\alpha(e_{1}))\eta^{2}$
(22)

while also (rewrite (12) using (20), apply $d\alpha$ to both sides of each
equation, and simplify)

$\sin\alpha\,d\alpha(e_{2})=\frac{1}{2}\left(-\frac{\partial\alpha}{\partial
u^{1}}+\frac{\partial\alpha}{\partial u^{2}}\right)\textrm{ and
}\cos\alpha\,d\alpha(e_{1})=\frac{1}{2}\left(\frac{\partial\alpha}{\partial
u^{1}}+\frac{\partial\alpha}{\partial u^{2}}\right).$

We define $\Theta:=2\alpha$, the angle between $E_{1}$ and $E_{2}$ (using the
capital letter to avoid confusion with the 1-forms $\theta^{1},\theta^{2}$).
Now (22) becomes

$\omega_{1}^{2}=-\frac{\partial\alpha}{\partial
u^{1}}\eta^{1}+\frac{\partial\alpha}{\partial
u^{2}}\eta^{2}=-\frac{1}{2}\frac{\partial\Theta}{\partial
u^{1}}\eta^{1}+\frac{1}{2}\frac{\partial\Theta}{\partial u^{2}}\eta^{2},$ (23)

which means that (using (16) and $\frac{1}{2}\sec\alpha\csc\alpha=\csc\Theta$
to show the final equality)

$\displaystyle d\omega_{1}^{2}$
$\displaystyle=-\frac{1}{2}\frac{\partial^{2}\Theta}{\partial u^{2}\partial
u^{1}}\eta^{2}\wedge\eta^{1}+\frac{1}{2}\frac{\partial^{2}\Theta}{\partial
u^{1}\partial u^{2}}\eta^{1}\wedge\eta^{2}=\frac{\partial^{2}\Theta}{\partial
u^{1}\partial u^{2}}(\eta^{1}\wedge\eta^{2})$
$\displaystyle=\csc\Theta\frac{\partial^{2}\Theta}{\partial u^{1}\partial
u^{2}}(\theta^{1}\wedge\theta^{2}),$ (24)

where $\theta^{1}\wedge\theta^{2}$ represents the “area form” $dA$ on
$\mathbb{D}$ (since $\\{e_{1},e_{2}\\}$ is orthonormal). At the same time,

$d\omega_{1}^{2}\overset{\eqref{Equ:dconn}}{=}-\omega_{1}^{3}\wedge\omega_{2}^{3}\overset{\eqref{Equ:princurv}}{=}-(\kappa_{1}\theta^{1})\wedge(\kappa_{2}\theta^{2})=-\kappa_{1}\kappa_{2}(\theta^{1}\wedge\theta^{2})=\theta^{1}\wedge\theta^{2}.$
(25)

Comparing equations (2.4) and (25), we have the PDE

$\frac{\partial^{2}\Theta}{\partial u^{1}\partial u^{2}}=\sin\Theta,$ (26)

on $\mathbb{D}$, which we can transfer to the $x^{1}x^{2}$-plane, after
defining $\hat{\Theta}:=\Theta\circ F^{-1}$, as

$\frac{\partial^{2}\hat{\Theta}}{\partial x^{1}\partial
x^{2}}=\sin\hat{\Theta}.$ (27)

By Proposition 3 and the definition of $\Theta$,
$\det(M_{\alpha})\overset{\eqref{Equ:GJacob}}{=}\sin(2\alpha)=\sin\Theta$ is
the factor by which area is distorted, when $F^{-1}$ maps the
$x^{1}x^{2}$-plane to $\mathbb{D}$. Hence, the area $A$ of the hyperbolic
plane is given by

$\displaystyle A$
$\displaystyle=\lim_{a\to\infty}\int_{-a}^{a}\int_{-a}^{a}\sin\hat{\Theta}\,dx^{1}\,dx^{2}=\lim_{a\to\infty}\int_{-a}^{a}\int_{-a}^{a}\frac{\partial^{2}\hat{\Theta}}{\partial
x^{1}\partial x^{2}}\,dx^{1}\,dx^{2}$
$\displaystyle=\int_{-a}^{a}\left[\frac{\partial\hat{\Theta}}{\partial
x^{2}}\right]_{x^{1}=-a}^{x^{1}=a}\,dx^{2}=\int_{-a}^{a}[\frac{\partial\hat{\Theta}}{\partial
x^{2}}(a,x^{2})-\frac{\partial\hat{\Theta}}{\partial
x^{2}}(-a,x^{2})]\,dx^{2}$
$\displaystyle=\lim_{a\to\infty}[\hat{\Theta}(a,a)-\hat{\Theta}(-a,a)-\hat{\Theta}(a,-a)+\hat{\Theta}(-a,-a)],$
(28)

and since $0<\Theta<\pi$, $A$ is bounded above by $2\pi$. On the other hand,
the hyperbolic plane has infinite area, as computed below (in the Poincaré
model; see the Appendix for the Riemannian metric):

$\displaystyle\lim_{t\to
1^{-}}\int_{0}^{2\pi}\int_{0}^{t}\frac{4}{(1-r^{2})^{2}}\,r\,dr\,d\theta$
$\displaystyle=\lim_{t\to
1^{-}}\int_{0}^{2\pi}\left[\frac{2}{(1-r^{2})}\right]_{0}^{t}\,d\theta$
$\displaystyle=2\pi\cdot\lim_{t\to
1^{-}}\left(\frac{2}{1-t^{2}}-2\right)=\infty$ (29)

Assuming the existence of $\phi$ has led to a contradiction, hence there does
not exist an isometric immersion of $\mathbb{D}$ into $\mathbb{R}^{3}$,
concluding the proof of Theorem 1. ∎

######  Corollary 3.

If $S$ is a $C^{\infty}$ Riemannian $2$-manifold for which the metric is
complete and has constant curvature $-1$, then there does not exist a
$C^{\infty}$ isometric immersion of $S$ into $\mathbb{R}^{3}$.

###### Proof.

By [Wol77, Corollary 2.3.17], the universal cover, $\tilde{S}$, of any such
$S$ would be isometric to $\mathbb{H}^{2}$, and the composition
$\tilde{S}\rightarrow S\rightarrow\mathbb{R}^{3}$ would be a counterexample to
Theorem 1. ∎

######  Corollary 4.

If $S$ is a $C^{\infty}$ Riemannian $2$-manifold for which the metric is
complete and has constant curvature $-k^{2}$ ($k>0$), then there does not
exist a $C^{\infty}$ isometric immersion of $S$ into $\mathbb{R}^{3}$.

###### Proof.

Suppose that $\psi:S\rightarrow\mathbb{R}^{3}$ is such an isometric immersion,
and let $h:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ be the dilation
$(x,y,z)\mapsto(kx,ky,kz)$. Since the effect of the dilation is to divide all
normal curvatures by $k$, the image of the composition $h\circ\psi$ has
constant curvature $-1$ everywhere, which contradicts Corollary 3 (if the
original inner product at every point on $S$ is multiplied by $k^{2}$, the new
Riemannian metric will still be complete, and $h\circ\psi$ will be an
isometric immersion). ∎

##  3 Brief History

Hilbert’s original proof [Hil01] assumed that the surface in $\mathbb{R}^{3}$
was real-analytic (and had constant negative curvature $-1$). However, the
outline of his argument will work if $\phi$ is smooth enough to ensure that
the PDE in equation (26) makes sense. Tracing back the definition of $\Theta$,
we can see that if $\phi\in C^{4}(\mathbb{D})$, then the normal vector
$\bar{e}_{3}$ is $C^{3}$, and the eigenvalues
$\bar{\kappa_{1}},\bar{\kappa}_{2}$ of $-d\bar{e}_{3}$ are $C^{2}$, as is
$\Theta=2\arctan(\kappa_{1})$. So $\partial^{2}\Theta/\partial u^{1}\partial
u^{2}$ exists and is continuous on $\mathbb{D}$. With more sophisticated
techniques, Efimov proved that there is no $C^{2}$ isometric immersion of the
hyperbolic plane; see [Mil72] for an exposition and more history. On the other
hand, by work of Nash and Kuiper, there _does_ exist a $C^{1}$ isometric
embedding of the hyperbolic plane in $\mathbb{R}^{3}$ [Nas54] [Kui55].

##  4 Appendix: Metric properties of $\mathbb{H}^{2}$

Let $O:=(0,0)\in\mathbb{D}:=\\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1\\}$.
$\mathbb{D}$ is the Poincaré disk model for $\mathbb{H}^{2}$, with Riemannian
metric $g_{11}=g_{22}=4/(1-x^{2}-y^{2})^{2},\ g_{12}=g_{21}=0$ [MP77, page
179], [Rat94, Theorem 4.5.5]. $\mathbb{D}$ inherits an orientation from the
usual orientation on $\mathbb{R}^{2}$. Furthermore, the corresponding distance
function on $\mathbb{D}$ is [Rat94, Theorem 4.5.1]

$d_{\mathbb{D}}(P,Q):=\operatorname{arccosh}\left(1+\frac{2|P-Q|^{2}}{(1-|P|^{2})(1-|Q|^{2})}\right)$

where $P,Q\in\mathbb{D}$ and $|P|$ denotes the Euclidean length of the vector
$P$.

Proofs of the following properties can be found in many textbooks, including
[Rat94].

  1. 1.

The metrics $g$ and $d_{\mathbb{D}}$ are invariant under Möbius
transformations which leave $\mathbb{D}$ invariant.

  2. 2.

The $d_{\mathbb{D}}$-metric topology on $\mathbb{D}$ equals the topology that
the $\mathbb{D}$ inherits as a subset of $\mathbb{R}^{2}$ (with the usual
topology).

  3. 3.

Every open $d_{\mathbb{D}}$-ball centered at a point in $\mathbb{D}$ equals
(as a set) an open Euclidean disk whose closure is contained in $\mathbb{D}$.

  4. 4.

The metric space $(\mathbb{D},d_{\mathbb{D}})$ is complete [Rat94, Theorem
8.5.1].

  5. 5.

After identifying $(x,y)$ with the complex number $z=x+iy$, the maps

$z\mapsto\frac{z-P}{-\bar{P}z+1},\ z\mapsto\frac{z+P}{\bar{P}z+1}$ (30)

are isometries of the Riemannian manifold $(\mathbb{D},g)$ and inverses of
each other, taking $P\in\mathbb{D}$ to $O$ and $O$ to $P$, respectively.
[Rat94, Section 4.5, Exercise 10] [Nee97, 3 IX & 6 III 11].

Furthermore, if $r>0$ and $\gamma:[0,r)\rightarrow\mathbb{D}$ is uniformly
continuous, then there is a unique extension of $\gamma$ to a (uniformly)
continuous map $\bar{\gamma}:[0,r]\rightarrow\mathbb{D}$. This is a very
special case of an exercise in [Mun00, Chapter 7]. The proof hinges on the
fact that every sequence in $[0,r)$ converging to $r$ is a Cauchy sequence and
will be mapped by $\gamma$ to a Cauchy sequence in $\mathbb{D}$.

## References

  * [BL10] Thomas Banchoff and Stephen Lovett, _Differential geometry of curves and surfaces_ , A K Peters, Ltd., Natick, MA, 2010. MR 2674651
  * [Cod61] Earl A. Coddington, _An introduction to ordinary differential equations_ , Prentice-Hall Mathematics Series, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961. MR 0126573
  * [dC83] Manfredo Perdigão do Carmo, _Formas diferenciais e aplicações_ , Monografías de Matemática [Mathematical Monographs], vol. 37, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1983, With an appendix by Rubens Leão. MR 752287
  * [dC94] Manfredo P. do Carmo, _Differential forms and applications_ , Universitext, Springer-Verlag, Berlin, 1994, Translated from the 1971 Portuguese original. MR 1301070
  * [dC16] , _Differential geometry of curves & surfaces_, Dover Publications, Inc., Mineola, NY, 2016, Revised & updated second edition of [MR0394451]. MR 3837152
  * [Hil01] David Hilbert, _Ueber Flächen von constanter Gaussscher Krümmung_ , Trans. Amer. Math. Soc. 2 (1901), no. 1, 87–99. MR 1500557
  * [Kui55] Nicolaas H. Kuiper, _On $C^{1}$-isometric imbeddings. I, II_, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 545–556, 683–689. MR 0075640
  * [Lee13] John M. Lee, _Introduction to smooth manifolds_ , second ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR 2954043
  * [Lim03] Elon Lages Lima, _Fundamental groups and covering spaces_ , A K Peters, Ltd., Natick, MA, 2003, Translated from the Portuguese by Jonas Gomes. MR 2000701
  * [Mil72] Tilla Klotz Milnor, _Efimov’s theorem about complete immersed surfaces of negative curvature_ , Advances in Math. 8 (1972), 474–543. MR 301679
  * [MP77] Richard S. Millman and George D. Parker, _Elements of differential geometry_ , Prentice-Hall Inc., Englewood Cliffs, N. J., 1977. MR 0442832
  * [Mun00] James R. Munkres, _Topology_ , Prentice Hall, Inc., Upper Saddle River, NJ, 2000, Second edition of [MR0464128]. MR 3728284
  * [Nas54] John Nash, _$C^{1}$ isometric imbeddings_, Ann. of Math. (2) 60 (1954), 383–396. MR 65993
  * [Nee97] Tristan Needham, _Visual complex analysis_ , The Clarendon Press, Oxford University Press, New York, 1997. MR 1446490
  * [O’N06] Barrett O’Neill, _Elementary differential geometry_ , second ed., Elsevier/Academic Press, Amsterdam, 2006. MR 2351345
  * [Rat94] John G. Ratcliffe, _Foundations of hyperbolic manifolds_ , Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730
  * [Shi18] Theodore Shifrin, _Differential geometry: a first course in curves and surfaces_ , https://math.franklin.uga.edu/sites/default/files/inline-files/ShifrinDiffGeo.pdf, 2018, preliminary version, accessed March 16, 2021.
  * [Spi79a] Michael Spivak, _A comprehensive introduction to differential geometry. Vol. I_ , second ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532830
  * [Spi79b] , _A comprehensive introduction to differential geometry. Vol. II_ , second ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532831
  * [Wol77] Joseph A. Wolf, _Spaces of constant curvature_ , fourth ed., Publish or Perish, Inc., Houston, TX, 1977. MR 928600