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
be connected, complete Riemannian manifolds. We consider the problem of isometrically mapping a small patch on
to a small patch on
.
Let
, and let
![{\displaystyle I:T_{x}M\rightarrow T_{y}N}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
be a linear isometry. This can be interpreted as isometrically mapping an infinitesimal patch (the tangent space) at
to an infinitesimal patch at
. Now we attempt to extend it to a finite (rather than infinitesimal) patch.
For sufficiently small
, 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}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
are local diffeomorphisms. Here,
is the ball centered on
of radius
One then defines a diffeomorphism
by
![{\displaystyle f=\exp _{y}\circ I\circ \exp _{x}^{-1}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
When is
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
, that is, it is an isometry on the infinitesimal patches. - It preserves the curvature tensor at the tangent space of every point on
, that is, it preserves how the infinitesimal patches fit together.
If
is an isometry, it must preserve the geodesics. Thus, it is natural to consider the behavior of
as we transport it along an arbitrary geodesic radius
starting at
. By property of the exponential mapping,
maps it to a geodesic radius of
starting at
,.
Let
be the parallel transport along
(defined by the Levi-Civita connection), and
be the parallel transport along
, 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]}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Cartan's theorem
The original theorem proven by Cartan is the local version of the Cartan–Ambrose–Hicks theorem.
is an isometry if and only if for all geodesic radii
with
, and all
, we have
where
are Riemann curvature tensors of
.
In words, it states that
is an isometry if and only if the only way to preserve its infinitesimal isometry also preserves the Riemannian curvature.
Note that
generally does not have to be a diffeomorphism, but only a locally isometric covering map. However,
must be a global isometry if
is simply connected.
Cartan–Ambrose–Hicks theorem
Theorem: For Riemann curvature tensors
and all broken geodesics (a broken geodesic is a curve that is piecewise geodesic)
with
, suppose that
![{\displaystyle I_{\gamma }(t)(R(X,Y,Z))={\overline {R}}(I_{\gamma }(t)(X),I_{\gamma }(t)(Y),I_{\gamma }(t)(Z))}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for all
.
Then, if two broken geodesics beginning at
have the same endpoint, the corresponding broken geodesics (mapped by
) in
also have the same end point. Consequently, there exists a map
defined
by mapping the broken geodesic endpoints in
to the corresponding geodesic endpoints in
.
The map
is a locally isometric covering map.
If
is also simply connected, then
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.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
A simply connected Riemannian manifold is locally symmetric if it is a symmetric space.
From the Cartan–Ambrose–Hicks theorem, we have:
Theorem: Let
be connected, complete, locally symmetric Riemannian manifolds, and let
be simply connected. Let their Riemann curvature tensors be
. Let
and
![{\displaystyle I:T_{x}M\rightarrow T_{y}N}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
be a linear isometry with
. Then there exists a locally isometric covering map
![{\displaystyle F:M\rightarrow N}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with
and
.
Corollary: Any complete locally symmetric space is of the form
, where
is a symmetric space and
is a discrete subgroup of isometries of
.
Classification of space forms
As an application of the Cartan–Ambrose–Hicks theorem, any simply connected, complete Riemannian manifold with constant sectional curvature
is respectively isometric to the n-sphere
, the n-Euclidean space
, and the n-hyperbolic space
.
References
- ^ Mathematics Genealogy Project, entry for Noel Justin Hicks
- ^ Ambrose, W. (1956). "Parallel Translation of Riemannian Curvature". The Annals of Mathematics. 64 (2). JSTOR: 337. doi:10.2307/1969978. ISSN 0003-486X.
- ^ Hicks, Noel (1959). "A theorem on affine connexions". Illinois Journal of Mathematics. 3 (2): 242–254. doi:10.1215/ijm/1255455125. ISSN 0019-2082.
- ^ Cheeger, Jeff; Ebin, David G. (2008). "Chapter 1, Section 12, The Cartan–Ambrose–Hicks Theorem". Comparison theorems in Riemannian geometry. Providence, R.I: AMS Chelsea Pub. ISBN 0-8218-4417-2. OCLC 185095562.