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Roughly speaking, a tangent vector is an infinitesimal displacement at a specific point on a manifold. The set of tangent vectors at a point P forms a vector space called the ...

A quaternion Kähler manifold is a Riemannian manifold of dimension 4n, n>=2, whose holonomy is, up to conjugacy, a subgroup of Sp(n)Sp(1)=Sp(n)×Sp(1)/Z_2, but is not a ...

A hyper-Kähler manifold can be defined as a Riemannian manifold of dimension 4n with three covariantly constant orthogonal automorphisms I, J, K of the tangent bundle which ...

A nonsimply connected 3-manifold, also called a dodecahedral space.

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 ...

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 ...

The space of immersions of a manifold in another manifold is homotopically equivalent to the space of bundle injections from the tangent space of the first to the tangent ...

A generalization of calculus of variations which draws the relationship between the stationary points of a smooth real-valued function on a manifold and the global topology ...

A complex vector bundle is a vector bundle pi:E->M whose fiber bundle pi^(-1)(x) is a complex vector space. It is not necessarily a complex manifold, even if its base ...

The canonical generator of the nonvanishing homology group on a topological manifold.