Glossary of Riemannian and Metric Geometry - C

C

Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to Rn via the exponential map; for metric spaces, the statement that a connected, simply connected complete geodesic metric space with non-positive curvature in the sense of Alexandrov is a (globally) CAT(0) space.

Cartan extended Einstein's General relativity to Einstein-Cartan theory, using Riemannian-Cartan geometry instead of Riemannian geometry. This extension provides affine torsion, which allows for non-symmetric curvature tensors and the incorporation of spin-orbit coupling.

Center of mass. A point q ∈ M is called the center of mass of the points if it is a point of global minimum of the function

Such a point is unique if all distances are less than radius of convexity.

Christoffel symbol

Collapsing manifold

Complete space

Completion

Conformal map is a map which preserves angles.

Conformally flat a M is conformally flat if it is locally conformally equivalent to a Euclidean space, for example standard sphere is conformally flat.

Conjugate points two points p and q on a geodesic are called conjugate if there is a Jacobi field on which has a zero at p and q.

Convex function. A function f on a Riemannian manifold is a convex if for any geodesic the function is convex. A function f is called -convex if for any geodesic with natural parameter, the function is convex.

Convex A subset K of a Riemannian manifold M is called convex if for any two points in K there is a shortest path connecting them which lies entirely in K, see also totally convex.

Cotangent bundle

Covariant derivative

Cut locus