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A manifold possessing a metric tensor. For a complete Riemannian manifold, the metric d(x,y) is defined as the length of the shortest curve (geodesic) between x and y. Every ...

The base manifold in a bundle is analogous to the domain for a set of functions. In fact, a bundle, by definition, comes with a map to the base manifold, often called pi or ...

A complex manifold is a manifold M whose coordinate charts are open subsets of C^n and the transition functions between charts are holomorphic functions. Naturally, a complex ...

An open three-manifold which is simply connected but is topologically distinct from Euclidean three-space.

A pair (M,omega), where M is a manifold and omega is a symplectic form on M. The phase space R^(2n)=R^n×R^n is a symplectic manifold. Near every point on a symplectic ...

An abstract manifold is a manifold in the context of an abstract space with no particular embedding, or representation in mind. It is a topological space with an atlas of ...

A subset M of a Hilbert space H is a linear manifold if it is closed under addition of vectors and scalar multiplication.

A topological space M satisfying some separability (i.e., it is a T2-space) and countability (i.e., it is a paracompact space) conditions such that every point p in M has a ...

A semi-Riemannian manifold M=(M,g) is said to be Lorentzian if dim(M)>=2 and if the index I=I_g associated with the metric tensor g satisfies I=1. Alternatively, a smooth ...

Two open manifolds M and M^' are cobordant if there exists a manifold with boundary W^(n+1) such that an acceptable restrictive relationship holds.