Search Results for ""

A pseudo-Riemannian manifold is a manifold which has a metric that is of the signature diag(-,+,...,+), as compared to a Riemannian manifold, which has a signature of all ...

In a complete metric space, a countable union of nowhere dense sets is said to be meager; the complement of such a set is a residual set.

A complex manifold for which the exterior derivative of the fundamental form Omega associated with the given Hermitian metric vanishes, so dOmega=0. In other words, it is a ...

A metric space X which is not complete has a Cauchy sequence which does not converge. The completion of X is obtained by adding the limits to the Cauchy sequences. For ...

A smooth manifold M=(M,g) is said to be semi-Riemannian if the indexMetric Tensor Index of g is nonzero. Alternatively, a smooth manifold is semi-Riemannian provided that it ...

A topological space X in which each subset of X of the "first category" has an empty interior. A topological space which is homeomorphic to a complete metric space is a Baire ...

If it is possible to transform a coordinate system to a form where the metric elements g_(munu) are constants independent of x^mu, then the space is flat.

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.

The indices of a contravariant tensor A^j can be lowered, turning it into a covariant tensor A_i, by multiplication by a so-called metric tensor g_(ij), e.g., g_(ij)A^j=A_i.

The indices of a covariant tensor A_j can be raised, forming a contravariant tensor A^i, by multiplication by a so-called metric tensor g^(ij), e.g., g^(ij)A_j=A^i