In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It is therefore a

tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...

of order three. It vanishes for the case of

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 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.

Relation to connection

We say that a connection

\Gamma

is compatible with the metric when its associated covariant derivative of the metric tensor (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, which is the only one without

torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...

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 In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...

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