In mathematics, the Cartan–Hadamard theorem is a statement in
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
concerning the structure of complete
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s of non-positive
sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a po ...
. The theorem states that the
universal cover
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphism ...
of such a manifold is
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Defini ...
to a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
via the
exponential map at any point. It was first proved by
Hans Carl Friedrich von Mangoldt for
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s in 1881, and independently by
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations.
Biography
The son of a tea ...
in 1898.
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...
generalized the theorem to Riemannian manifolds in 1928 (; ; ). The theorem was further generalized to a wide class of
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s by
Mikhail Gromov in 1987; detailed proofs were published by for metric spaces of non-positive curvature and by for general locally convex metric spaces.
Riemannian geometry
The Cartan–Hadamard theorem in conventional Riemannian geometry asserts that the
universal covering space of a
connected complete Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
of non-positive
sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a po ...
is
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Defini ...
to R
''n''. In fact, for complete manifolds of non-positive curvature, the
exponential map based at any point of the manifold is a covering map.
The theorem holds also for
Hilbert manifolds in the sense that the exponential map of a non-positively curved geodesically complete connected manifold is a covering map (; ). Completeness here is understood in the sense that the exponential map is defined on the whole
tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
of a point.
Metric geometry
In
metric geometry
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, the Cartan–Hadamard theorem is the statement that the universal cover of a
connected non-positively curved complete metric space ''X'' is a
Hadamard space. In particular, if ''X'' is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
then it is a geodesic space in the sense that any two points are connected by a unique minimizing geodesic, and hence
contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within t ...
.
A metric space ''X'' is said to be non-positively curved if every point ''p'' has a neighborhood ''U'' in which any two points are joined by a
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
, and for any point ''z'' in ''U'' and constant speed geodesic γ in ''U'', one has
:
This inequality may be usefully thought of in terms of a
geodesic triangle
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conne ...
Δ = ''z''γ(0)γ(1). The left-hand side is the square distance from the vertex ''z'' to the midpoint of the opposite side. The right-hand side represents the square distance from the vertex to the midpoint of the opposite side in a Euclidean triangle having the same side lengths as Δ. This condition, called the
CAT(0) condition is an abstract form of
Toponogov's triangle comparison theorem.
Generalization to locally convex spaces
The assumption of non-positive curvature can be weakened , although with a correspondingly weaker conclusion. Call a metric space ''X'' convex if, for any two constant speed minimizing geodesics ''a''(''t'') and ''b''(''t''), the function
:
is a
convex function
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...
of ''t''. A metric space is then locally convex if every point has a neighborhood that is convex in this sense. The Cartan–Hadamard theorem for locally convex spaces states:
* If ''X'' is a locally convex complete connected metric space, then the universal cover of ''X'' is a convex geodesic space with respect to the
induced length metric ''d''.
In particular, the universal covering of such a space is contractible. The convexity of the distance function along a pair of geodesics is a well-known consequence of non-positive curvature of a metric space, but it is not equivalent .
Significance
The Cartan–Hadamard theorem provides an example of a local-to-global correspondence in Riemannian and metric geometry: namely, a local condition (non-positive curvature) and a global condition (simple-connectedness) together imply a strong global property (contractibility); or in the Riemannian case, diffeomorphism with R
n.
The metric form of the theorem demonstrates that a non-positively curved polyhedral cell complex is
aspherical. This fact is of crucial importance for modern
geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these group ...
.
See also
*
Glossary of Riemannian and metric geometry
*
Cartan–Hadamard manifold
*
Cartan–Hadamard conjecture
References
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{{DEFAULTSORT:Cartan-Hadamard theorem
Metric geometry
Theorems in Riemannian geometry