Riemann curvature tensor in the context of "Sectional curvature"

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects.The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.

In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, for example, the scalar product of vectors, or by contracting a tensor. While the components of the contracted quantities may change under Lorentz transformations, the Lorentz scalars remain unchanged.

A simple Lorentz scalar in Minkowski spacetime is the spacetime distance ("length" of their difference) of two fixed events in spacetime. While the "position"-4-vectors of the events change between different inertial frames, their spacetime distance remains invariant under the corresponding Lorentz transformation. Other examples of Lorentz scalars are the "length" of a 4-velocity (see below), or the Ricci curvature at a point in spacetime in general relativity, which is a contraction of the Riemann curvature tensor.

In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in material object, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a scalar field and a vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor A is defined on a vector fields set X(M) over a module M, we call A a tensor field on M.A tensor field, in common usage, is often referred to in the shorter form "tensor". For example, the Riemann curvature tensor refers a tensor field, as it associates a tensor to each point of a Riemannian manifold, a topological space.

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.

Tensors have become important in physics, because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, etc.), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, etc.), and general relativity (stress–energy tensor, curvature tensor, etc.). In applications, it is common to study situations in which a different tensor can occur at each point of an object. For example, the stress within an object may vary from one location to another. A family of tensors, that vary across space in this way, is a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".