Cartan–Ambrose–Hicks theorem

Let ${\displaystyle M,N}$ be connected, complete Riemannian manifolds. We consider the problem of isometrically mapping a small patch on ${\displaystyle M}$ to a small patch on ${\displaystyle N}$.

Let ${\displaystyle x\in M,y\in N}$, and let

${\displaystyle I:T_{x}M\rightarrow T_{y}N}$

be a linear isometry. This can be interpreted as isometrically mapping an infinitesimal patch (the tangent space) at ${\displaystyle x}$ to an infinitesimal patch at ${\displaystyle y}$. Now we attempt to extend it to a finite (rather than infinitesimal) patch.

For sufficiently small ${\displaystyle r>0}$, the exponential maps

${\displaystyle \exp _{x}:B_{r}(x)\subset T_{x}M\rightarrow M,\exp _{y}:B_{r}(y)\subset T_{y}N\rightarrow N}$

are local diffeomorphisms. Here, ${\displaystyle B_{r}(x)}$ is the ball centered on ${\displaystyle x}$ of radius ${\displaystyle r.}$ One then defines a diffeomorphism ${\displaystyle f:B_{r}(x)\rightarrow B_{r}(y)}$ by

${\displaystyle f=\exp _{y}\circ I\circ \exp _{x}^{-1}.}$

When is ${\displaystyle f}$ an isometry? Intuitively, it should be an isometry if it satisfies the two conditions:

• It is a linear isometry at the tangent space of every point on ${\displaystyle B_{r}(x)}$, that is, it is an isometry on the infinitesimal patches.
• It preserves the curvature tensor at the tangent space of every point on ${\displaystyle B_{r}(x)}$, that is, it preserves how the infinitesimal patches fit together.

If ${\displaystyle f}$ is an isometry, it must preserve the geodesics. Thus, it is natural to consider the behavior of ${\displaystyle f}$ as we transport it along an arbitrary geodesic radius ${\displaystyle \gamma$ :\left[0,T\right]\rightarrow B_{r}(x)\subset M} starting at ${\displaystyle \gamma (0)=x}$. By property of the exponential mapping, ${\displaystyle f}$ maps it to a geodesic radius of ${\displaystyle B_{r}(y)}$ starting at ${\displaystyle f(\gamma )(0)=y}$,.

Let ${\displaystyle P_{\gamma }(t)}$ be the parallel transport along ${\displaystyle \gamma }$ (defined by the Levi-Civita connection), and ${\displaystyle P_{f(\gamma )(t)}}$ be the parallel transport along ${\displaystyle f(\gamma )}$, then we have the mapping between infinitesimal patches along the two geodesic radii:

${\displaystyle I_{\gamma }(t)=P_{f(\gamma )(t)}\circ I\circ P_{\gamma (t)}^{-1}:T_{\gamma (t)}M\rightarrow T_{f(\gamma (t))}N\quad {\text{ for all }}t\in [0,T]}$

The original theorem proven by Cartan is the local version of the Cartan–Ambrose–Hicks theorem.

${\displaystyle f}$ is an isometry if and only if for all geodesic radii ${\displaystyle \gamma$ :\left[0,T\right]\rightarrow B_{r}(x)\subset M} with ${\displaystyle \gamma (0)=x}$, and all ${\displaystyle t\in [0,T],X,Y,Z\in T_{\gamma (t)}M}$, we have ${\displaystyle I_{\gamma }(t)(R(X,Y,Z))={\overline {R}}(I_{\gamma }(t)(X),I_{\gamma }(t)(Y),I_{\gamma }(t)(Z))}$ where ${\displaystyle R,{\overline {R}}}$ are Riemann curvature tensors of ${\displaystyle M,N}$.

In words, it states that ${\displaystyle f}$ is an isometry if and only if the only way to preserve its infinitesimal isometry also preserves the Riemannian curvature.

Note that ${\displaystyle f}$ generally does not have to be a diffeomorphism, but only a locally isometric covering map. However, ${\displaystyle f}$ must be a global isometry if ${\displaystyle N}$ is simply connected.

Theorem: For Riemann curvature tensors ${\displaystyle R,{\overline {R}}}$ and all broken geodesics (a broken geodesic is a curve that is piecewise geodesic) ${\displaystyle \gamma$ :\left[0,T\right]\rightarrow M} with ${\displaystyle \gamma (0)=x}$,

${\displaystyle I_{\gamma }(t)(R(X,Y,Z))={\overline {R}}(I_{\gamma }(t)(X),I_{\gamma }(t)(Y),I_{\gamma }(t)(Z))}$

for all ${\displaystyle t\in [0,T],X,Y,Z\in T_{\gamma (t)}M}$.

Then, if two broken geodesics beginning in ${\displaystyle x}$ have the same endpoint, then the corresponding broken geodesics (mapped by ${\displaystyle I_{\gamma }}$) in ${\displaystyle N}$ also have the same end point. So there exists a map

${\displaystyle F:M\rightarrow N}$

by mapping the broken geodesic endpoints in ${\displaystyle M}$to the corresponding geodesic endpoints in ${\displaystyle N}$.

The map ${\displaystyle F:M\rightarrow N}$ is a locally isometric covering map.

If ${\displaystyle N}$ is also simply connected, then ${\displaystyle F}$ is an isometry.

A Riemannian manifold is called locally symmetric if its Riemann curvature tensor is invariant under parallel transport:

${\displaystyle \nabla R=0.}$

A simply connected Riemannian manifold is locally symmetric if it is a symmetric space.

From the Cartan–Ambrose–Hicks theorem, we have:

Theorem: Let ${\displaystyle M,N}$ be connected, complete, locally symmetric Riemannian manifolds, and let ${\displaystyle M}$ be simply connected. Let their Riemann curvature tensors be ${\displaystyle R,{\overline {R}}}$. Let ${\displaystyle x\in M,y\in N}$ and

${\displaystyle I:T_{x}M\rightarrow T_{y}N}$

be a linear isometry with ${\displaystyle I(R(X,Y,Z))={\overline {R}}(I(X),I(Y),I(Z))}$. Then there exists a locally isometric covering map

${\displaystyle F:M\rightarrow N}$

with ${\displaystyle F(x)=y}$ and ${\displaystyle D_{x}F=I}$.

Corollary: Any complete locally symmetric space is of the form ${\displaystyle M/\gamma }$ for a symmetric space ${\displaystyle M}$and ${\displaystyle \gamma \subset Isom(M)}$ is a discrete subgroup of isometries of ${\displaystyle M}$.

As an application of the Cartan–Ambrose–Hicks theorem, any simply connected, complete Riemannian manifold with constant sectional curvature ${\displaystyle \in \{+1,0,-1\}}$ is respectively isometric to the n-sphere ${\displaystyle S^{n}}$, the n-Euclidean space ${\displaystyle E^{n}}$, and the n-hyperbolic space ${\displaystyle \mathbb {H} ^{n}}$.