In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product...

Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free...

describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing...

Relativity, dynamic variables of metric-affine gravitation theory are both a pseudo-Riemannian metric and a general linear connection on a world manifold ⁠ X {\displaystyle...

the metric connection is called the Riemannian connection. Given a Riemannian connection, one can always find a unique, equivalent connection that is...

metric has non-zero curvature, and the circle is a geodesic, so that its field of tangent vectors is parallel. A metric connection is any connection whose...

Yang-Mills connection is a special metric connection which satisfies the Yang-Mills equations of motion. A Riemannian connection is a metric connection on the...

the metric h {\displaystyle h} rather than the associated Chern connection, and such metrics solving the equations are called Hermite–Einstein metrics. The...

Levi-Civita connection is a torsion-free connection that preserves the metric. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection. Note...

infinitely many affine connections. If the manifold is further endowed with a metric tensor then there is a natural choice of affine connection, called the Levi-Civita...

relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the...

Torsion-free affine connection, an affine connection whose torsion tensor vanishes Torsion-free metric connection or Levi-Civita connection, a unique symmetric...

sufficient to ask that a vector bundle have a metric connection; when one does so, one finds that the metric connection satisfies the Yang–Mills equations of...

tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute...

Vol II.). A Chern connection, a connection of a holomorphic vector bundle with a Hermitian structure, is the unique metric connection D, such that the...

geometric structures. In metric geometry, the contorsion tensor expresses the difference between a metric-compatible affine connection with Christoffel symbol...

are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion. Geodesics are of particular...

affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-)Riemannian connection of the given metric. Because...

the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics. Every smooth complex projective...

called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian...

provide more detailed expositions of the definitions given below. Connection Curvature Metric space Riemannian manifold See also: Glossary of general topology...

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field...

be represented with the same numbers. When using the metric connection (Levi-Civita connection), the covariant derivative of an even tensor density is...

In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface)...

pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs...

space M is also a metric space, with metric g, and the principal bundle is endowed with a connection form ω, then π*g+kω is a metric defined on the entire...

manifolds are also an interesting class of manifolds admitting a metric connection with parallel totally antisymmetric torsion. Nearly Kähler manifolds...

distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space M {\displaystyle M} . It...

especially in connection with the filling area conjecture in Riemannian geometry, but this term has also been used for other concepts. A metric circle, defined...

of the metric, although in general relativity one usually uses an expression that seemingly depends on the metric through the affine connection. Whereas...