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]

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 ${\displaystyle \gamma (0)=x}$, suppose that

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

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

Then, if two broken geodesics beginning at ${\displaystyle x}$ have the same endpoint, the corresponding broken geodesics (mapped by ${\displaystyle I_{\gamma }}$) in ${\displaystyle N}$ also have the same end point. Consequently, there exists a map ${\displaystyle F:M\rightarrow N}$ defined 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 ${\displaystyle M/\Gamma }$, where ${\displaystyle M}$ is a symmetric space and ${\displaystyle \Gamma \subset \mathrm {Isom} (M)}$ 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 ${\displaystyle \mathbb {H} ^{n}}$.