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The Riemannian connection or Levi-Civita connection [9] is perhaps most easily understood in terms of lifting vector fields, considered as first order differential operators acting on functions on the manifold, to differential operators on sections of the frame bundle. In the case of an embedded surface, this lift is very simply described in ...
In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ()Riemannian metric and is torsion-free.
A connection for which the covariant derivatives of the metric on E vanish. A principal connection on the bundle of orthonormal frames of E. A special case of a metric connection is a Riemannian connection; there exists a unique such connection which is torsion free, the Levi-Civita connection.
An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor, with any given vector-valued 2-form as its torsion. The difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the contorsion tensor.
Riemannian geometry, the study of Riemannian manifolds, has deep connections to other areas of math, including geometric topology, complex geometry, and algebraic geometry. Applications include physics (especially general relativity and gauge theory), computer graphics, machine learning, and cartography.
In pseudo-Riemannian and Riemannian geometry the Levi-Civita connection is a special connection associated to the metric tensor. These are examples of affine connections . There is also a concept of projective connection , of which the Schwarzian derivative in complex analysis is an instance.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume.
A Riemannian connection is a metric connection on the tangent bundle of a Riemannian manifold. A Levi-Civita connection is a special Riemannian connection: the metric-compatible connection on the tangent bundle that is also torsion-free. It is unique, in the sense that given any Riemannian connection, one can always find one and only one ...