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 po ...

and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...

. These tensors are usually the Riemann tensor, 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 forc ...

, 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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

contractions and covariant differentiations.

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 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 ele ...

s. However, it can also be considered a

linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...

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 can ...

s, and as such it has a

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

, 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 denot ...

s) are polynomial scalar invariants.

Physical applications

In metric theories of gravitation 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 In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann. Definition The Kretschmann invariant is : K ...

R_ \, R^

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

R_ \, ^

These are analogous to two familiar quadratic invariants of the

electromagnetic field tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...

in classical electromagnetism. An important unsolved problem in general relativity is to give a basis (and any

syzygies Syzygy (from Greek Συζυγία "conjunction, yoked together") may refer to: Science * Syzygy (astronomy), a collinear configuration of three celestial bodies * Syzygy (mathematics), linear relation between generators of a module * Syzygy, ...

) 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 Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...

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