curvature-of-metric

[flat-metric]

flat metric := a metric such that there exist coordinates where standard metric

The submanifold metric of inherited from is not flat-metric, but rather constant-sectional-curvature.

When does a flat metric exist?

Choose a coordinate , with metric

Assume transformation to with metric , then

Connection transformations

For flat metric , thus

Equivalently

This property, combined with the initial condition of , allows for the recovery of the flat metric using PDE, i.e., for proving

Proof

product-rule expansion of the above differential

linear PDE for

is solvable <==> satisfies linear-PDE-integrable-condition

where is geodesic-derivative

If is solved, integrating again with the initial condition yields , which gives the conversion function from coordinate to the flat-metric coordinate

In flat-metric coordinates , so the geodesic ODE is , so flat-metric coordinates will be geodesic coordinates

When not exist flat metric coordinate, choose Einstein-metric as minimal scalar-curvature

Now, do not assume flat metric

[curvature-of-metric]

Curvature ( from "Riemann")

is a tensor (even though is not)

name-overload: Curvature := metric-dual of curvature

In coordinates

[flat-metric-iff-curvature-0] flat-metric <==> curvature is zero

[curvature-determine-metric-locally]

"flat-metric <==> curvature zero" can be generalized to curvature determines local metric

If two metrics and their curvatures are related by a local diffeomorphism between points , and the differential is an isometry between the tangent spaces , then the local diffeomorphism is a local isometry of .

[curvature-in-geodesic-coordinate]

At the origin of geodesic coordinates, by calculation, through

metric-connection definition of and definition of curvature

we have

==> If in geodesic coordinates, the second order derivative of the Taylor expansion of the metric is also zero then the curvature is also zero , which leads to a flat-metric, and thus the higher order derivatives are also zero

[symmetry-of-curvature]

==>

Proof Definition of curvature in geodesic coordinates, expressed using or

[algebraic-curvature-tensor] The algebraic curvature tensor is defined to satisfy the above symmetries (ref-6, lect 8)

[curvature-product]

Mimicking the definition of curvature in geodesic coordinates, for second-order symmetric tensors , define the curvature-product

satisfies symmetry-of-curvature, so , or

At the origin of geodesic coordinates, the curvature is (formally)

Def

maps to itself and , i.e. wiki:Projection_(linear_algebra), so
For alternating tensors , , so

[dimension-of-algebraic-curvature] Using , we have the dimension of the algebraic curvature tensor space

where

metric is a tensor

Mapping

[adjoint-of-curvature-product] :=

For and and the tensor-induced-metric of

For each , the linear function

has the metric-adjoint of the space :=

The linear function can be represented by the metric of the space

In coordinates

is injective, and is surjective. Proof uses the pre-inverse and post-inverse of composite mappings. The construction method refers to the calculation in curvature-decomposition

metric-adjoint ==>

Linear mapping ==>

==>

Mapping

metric-adjoint :=

for and

In coordinates

is injective, is surjective

[curvature-decomposition-space]

Orthogonal decomposition into tensor subspace, and cannot be decomposed further, i.e., irreducible

[curvature-decomposition] forall , exists , orthogonal decomposition

Proof if it's true then

[Ricci-curvature]

In coordinates
[scalar-curvature]

In coordinates
[conformal-curvature] (named so because if it vanish then the metric conformally flat when ) ( from "Weyl")

Similarly, the orthogonal decomposition is

trace-free Ricci-curvature

Curvature Orthogonal Decomposition to Tensor Subspace

quadratic-form

[curvature-low-dimension] low dimension curvature

span by

, only type , and

is a bijection
is completely determined by

let

Curvature also appears in the Taylor expansion of metric geodesic coordinates

and satisfies

Note that this is an equality of sums , not an equality of coefficients

"trace" also appears in Taylor-expansion of metric volume-form , related to and
"trace" appears again in the volume of geodesic ball (for spacetime manifold should use multi radius?)

if scale matric

geodesic-derivative
curvature
curvature metric-dual
Ricci-curvature
sectional_curvature
scalar-curvature

When representing spacetime metric with signature , it is obtained by multiplying the signature by

note-math