Torsion and Curvature

Torsion

Torsion

Let be a manifold with an affine connection on the tangent bundle. The torsion tensor of is the vector valued -form defined on vector fields and by where is the Lie bracket of and .We say an affine connection is torsion free if its torsion tensor vanishes identically.

Proposition

The torsion is tensorial.

Proof Let be vector fields and a smooth function. Then we have $$\begin{aligned}T(fX,Y)&=\nabla_{fX}Y-\nabla_Y(fX)-[fX,Y]\&=f\nabla_XY- Y(f)X-f\nabla_YX+Y(f)X-f[X,Y]\&=f(\nabla_XY-\nabla_YX-[X,Y])\&=fT(X,Y)\end{aligned}$$$\square$

Proposition

In a local coordinate system, the torsion tensor is given by

Proof It follows that .

Corollary

An affine connection is torsion free if and only if the first two components of the Christoffel symbols are symmetric. i.e. .

Proof It is straightforward in coordinate form.

Curvature

Riemann Curvature

Let be a manifold with an affine connection on the tangent bundle. The (Riemann) curvature tensor of is the vector valued -form defined on vector fields and by Alternatively, the Riemann tensor can be defined as a -tensor that acts on and by

In fact, we can also introduce the curvature tensor to any vector bundle:

Curvature Tensor

Suppose is a connection on a vector bundle over , then the curvature tensor of is the section of defined by

Geometric Meaning of Curvature

The Riemann curvature tensor gives a measure of the non-commutativity of covariant derivatives acting on tensor fields on .

Proposition

An affine connection is flat if and only if there exist a local coordinate system in which Christoffel symbols vanish.

Proof

Theorem

An affine connection is flat if and only if the parallel transport is path independent.

Proposition

The components of the Riemann curvature tensor are given by

Ricci Tensor

Ricci tensor is the contraction of the Riemann curvature tensor over the first and third indices (equivalently, the second and fourth indices). Explicitly, it is a -tensor such that

Proposition

The Ricci tensor is symmetric.

Proof

We can further contract the Ricci tensor to obtain the Ricci scalar. But since the Ricci Tensor is already purely covariant, one cannot contract it further without introducing a metric. So we have the following definition:

Ricci Scaler

Ricci scaler is a -tensor, i.e. a smooth function, defined on a semi Riemannian manifold by