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The Betti numbers of a compact orientable n-manifold satisfy the relation b_i=b_(n-i).

A compact manifold admits a Lorentzian structure iff its Euler characteristic vanishes. Therefore, every noncompact manifold admits a Lorentzian structure.

The index of a vector field with finitely many zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold.

A C^infty (infinitely differentiable) manifold is said to be a submanifold of a C^infty manifold M^' if M is a subset of M^' and the identity map of M into M^' is an ...

An integer kappa equal to 0 or 1 which vanishes iff the product manifold M^4×R can be given a smooth structure. Here, M^n is a compact connected topological four-manifold.

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 ...

Any n-dimensional Riemannian manifold can be locally embedded into an (n+1)-dimensional manifold with Ricci curvature Tensor R_(ab)=0. A similar version of the theorem for a ...

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

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 ...

On a compact oriented Finsler manifold without boundary, every cohomology class has a unique harmonic representation. The dimension of the space of all harmonic forms of ...