Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

and

pseudo-Riemannian geometry In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which t ...

, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually the

Riemann tensor 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 common way used to express the curvature of Riemannian manifolds. ...

, the

Weyl tensor In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal f ...

, the

Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...

and tensors formed from these by the operations of taking dual

contraction Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...

s and

covariant differentiation In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differ ...

Types of curvature invariants

The invariants most often considered are ''polynomial invariants''. These are polynomials constructed from contractions such as traces. Second degree examples are called ''quadratic invariants'', and so forth. Invariants constructed using covariant derivatives up to order n are called n-th order ''differential invariants''. The Riemann tensor is a

multilinear operator In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...

of fourth rank acting on

tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...

s. However, it can also be considered a linear operator acting on

bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...

s, and as such it has a characteristic polynomial, whose coefficients and roots (

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...

s) are polynomial scalar invariants.

Physical applications

metric theories of gravitation Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...

such as

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

, curvature scalars play an important role in telling distinct spacetimes apart. Two of the most basic curvature invariants in general relativity are the Kretschmann scalar :

R_ \, R^

and the ''Chern–Pontryagin scalar'', :

R_ \, ^

These are analogous to two familiar quadratic invariants of the electromagnetic field tensor in classical electromagnetism. An important unsolved problem in general relativity is to give a

basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...

(and any syzygies) for the zero-th order invariants of the Riemann tensor. They have limitations because many distinct spacetimes cannot be distinguished on this basis. In particular, so called VSI spacetimes (including pp-waves as well as some other Petrov type N and III spacetimes) cannot be distinguished from

Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...

using any number of polynomial curvature invariants (of any order).

References

* Riemannian geometry {{differential-geometry-stub

Types of curvature invariants

Physical applications

See also

References