Cartan–Ambrose–Hicks theorem

In mathematics, the Cartan–Ambrose–Hicks theorem is a theorem of Riemannian geometry, according to which the Riemannian metric is locally determined by the Riemann curvature tensor, or in other words, behavior of the curvature tensor under parallel translation determines the metric.

The theorem is named after Élie Cartan, Warren Ambrose, and his PhD student Noel Hicks.^[1] Cartan proved the local version. Ambrose proved a global version that allows for isometries between general Riemannian manifolds with varying curvature, in 1956.^[2] This was further generalized by Hicks to general manifolds with affine connections in their tangent bundles, in 1959.^[3]

A statement and proof of the theorem can be found in ^[4]

Introduction[]

Let ${\displaystyle M,N}$ be connected, complete Riemannian manifolds. Let ${\displaystyle x\in M,y\in N}$, and let

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

be a linear isometry. 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

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

For a geodesic ${\displaystyle \gamma :\left[0,T\right]\rightarrow B_{r}(x)\subset M}$ with ${\displaystyle \gamma (0)=x}$, ${\displaystyle f}$ maps it to a geodesic ${\displaystyle f(\gamma ):\left[0,T\right]\rightarrow B_{r}(y)\subset N}$ with ${\displaystyle f(\gamma )(0)=y}$,. 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 ${\displaystyle f(\gamma )}$. We then define

${\displaystyle I_{\gamma }(t)=P_{f(\gamma )(t)}\circ I\circ P_{\gamma (t)}^{-1}:T_{\gamma (t)}M\rightarrow T_{f(\gamma (t))}N}$

for ${\displaystyle t\in \left[0,T\right]}$.

Cartan's theorem[]

The original theorem proven by Cartan is the local version of the Cartan–Ambrose–Hicks theorem. It states that ${\displaystyle f}$ is a (local) isometry if for all geodesics ${\displaystyle \gamma :\left[0,T\right]\rightarrow B_{r}(x)\subset M}$ 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 ${\displaystyle M,N}$.

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.

Cartan–Ambrose–Hicks theorem[]

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.

Locally symmetric spaces[]

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 ${\displaystyle M}$.

Classification of space forms[]

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 ${\displaystyle \mathbb {H} ^{n}}$.

References[]