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
be connected, complete Riemannian manifolds.
We consider the problem of isometrically mapping a small patch on
This can be interpreted as isometrically mapping an infinitesimal patch (the tangent space) at
Now we attempt to extend it to a finite (rather than infinitesimal) patch.
, the exponential maps are local diffeomorphisms.
Intuitively, it should be an isometry if it satisfies the two conditions: If
is an isometry, it must preserve the geodesics.
as we transport it along an arbitrary geodesic radius
(defined by the Levi-Civita connection), and
, then we have the mapping between infinitesimal patches along the two geodesic radii:
is an isometry if and only if for all geodesic radii
are Riemann curvature tensors of
is an isometry if and only if the only way to preserve its infinitesimal isometry also preserves the Riemannian curvature.
generally does not have to be a diffeomorphism, but only a locally isometric covering map.
Theorem: For Riemann curvature tensors
have the same endpoint, the corresponding broken geodesics (mapped by
defined by mapping the broken geodesic endpoints in
is a locally isometric covering map.
A Riemannian manifold is called locally symmetric if its Riemann curvature tensor is invariant under parallel transport: A simply connected Riemannian manifold is locally symmetric if it is a symmetric space.
be connected, complete, locally symmetric Riemannian manifolds, and let
Then there exists a locally isometric covering map with
Corollary: Any complete locally symmetric space is of the form
is a discrete subgroup of isometries of
As an application of the Cartan–Ambrose–Hicks theorem, any simply connected, complete Riemannian manifold with constant sectional curvature