In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most …

Chapter 14 Curvature in Riemannian Manifolds 14.1 The Curvature Tensor Since the notion of curvature can be defined for curves and surfaces, it is natural to wonder whether it can be …

There is another tensor, simply called the curvature tensor, that can be used to express the curvature of a surface. This tensor depends on the way in which the surface is embedded in …

Lecture 16. Curvature In this lecture we introduce the curvature tensor of a Riemannian manifold, and investigate its algebraic structure.

In two dimensions, it is twice the Gauss curvature. The Einstein tensor G is defined as R − Sg/2 . A metric is called an Einstein metric if R = λg for some λ.

Thus the Riemann curvature tensor is Rm(X, Y, Z, W ) = − Y, Z X, W + X, Z Y, W . Now we introduce the Kulkarni-Nomizu product ∧ that converts 2 symmetric (0, 2)-tensors T1 and T2 …

Nov 14, 2025 · The Riemann tensor (Schutz 1985) , also known the Riemann-Christoffel curvature tensor (Weinberg 1972, p. 133; Arfken 1985, p. 123) or Riemann curvature tensor …

Feb 23, 2010 · In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single …

In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a …

The Riemann curvature tensor that we derived in the previous lecture is one of the most important quantities we have developed this term. Certain variants of this curvature are also important.