mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the nonmetricity tensor in

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

is the

covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...

of the

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...

. It is therefore a

tensor field 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, ...

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

three. It vanishes for the case of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

and can be used to study non-Riemannian spacetimes.

Definition

By components, it is defined as follows. :

Q_=\nabla_g_

It measures the rate of change of the components of the metric tensor along the flow of a given vector field, since :

\nabla_\equiv\nabla_

where

\_

is the coordinate basis of vector fields of the tangent bundle, in the case of having a 4-dimensional

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

Relation to connection

We say that a

connection Connection may refer to: Mathematics *Connection (algebraic framework) *Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold * Connection (affine bundle) *Connection (composite bun ...

\Gamma

is compatible with the metric when its associated covariant derivative of the

(call it

\nabla^

, for example) is zero, i.e. :

\nabla^_g_=0 .

If the connection is also torsion-free (i.e. totally symmetric) then it is known as the

Levi-Civita connection 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 ( pseudo-) Riemannian ...

, which is the only one without torsion and compatible with the metric tensor. If we see it from a geometrical point of view, a non-vanishing nonmetricity tensor for a metric tensor

g

implies that the modulus of a vector defined on the

tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...

to a certain point

p

of the manifold, ''changes'' when it is evaluated along the direction (flow) of another arbitrary vector.

References

External links

* Differential geometry {{differential-geometry-stub