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On a Riemannian manifold, there is a unique connection which is torsion-free and compatible with the metric. This connection is called the Levi-Civita connection.

Homology is a concept that is used in many branches of algebra and topology. Historically, the term "homology" was first used in a topological sense by Poincaré. To him, it ...

A Kähler structure on a complex manifold M combines a Riemannian metric on the underlying real manifold with the complex structure. Such a structure brings together geometry ...

Topological lower bounds in terms of Betti numbers for the number of critical points form a smooth function on a smooth manifold.

A vector field is a section of its tangent bundle, meaning that to every point x in a manifold M, a vector X(x) in T_xM is associated, where T_x is the tangent space.

On a Lie group, exp is a map from the Lie algebra to its Lie group. If you think of the Lie algebra as the tangent space to the identity of the Lie group, exp(v) is defined ...

The cotangent bundle of a manifold is similar to the tangent bundle, except that it is the set (x,f) where x in M and f is a dual vector in the tangent space to x in M. The ...

A theorem which states that if a Kähler form represents an integral cohomology class on a compact manifold, then it must be a projective Abelian variety.

A relation between compact boundaryless manifolds (also called closed manifolds). Two closed manifolds are bordant iff their disjoint union is the boundary of a compact ...

A pullback is a general categorical operation appearing in a number of mathematical contexts, sometimes going under a different name. If T:V->W is a linear transformation ...