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Definition

Riemann curvature tensor

The Riemann curvature tensor measures the failure of Levi–Civita covariant derivatives to commute.

Definition

Let (M,g)(M,g) be a with ∇\nabla. Its Riemann curvature tensor is the (1,3)(1,3)-tensor

R(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]ZR(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z

for smooth X,Y,ZX,Y,Z. The metric-lowered version is the (0,4)(0,4)-tensor

Rm⁡(X,Y,Z,W)=g(R(X,Y)Z,W).\operatorname{Rm}(X,Y,Z,W)=g(R(X,Y)Z,W).

The bracket correction makes RR linear over smooth functions in all three arguments, so its value at a point depends only on the tangent vectors there. This convention fixes the overall sign of both tensors.

Algebraic symmetries

For the convention in the core, the lowered tensor satisfies

Rm⁡(X,Y,Z,W)=−Rm⁡(Y,X,Z,W)=−Rm⁡(X,Y,W,Z)\operatorname{Rm}(X,Y,Z,W)=-\operatorname{Rm}(Y,X,Z,W) =-\operatorname{Rm}(X,Y,W,Z)

and

Rm⁡(X,Y,Z,W)=Rm⁡(Z,W,X,Y).\operatorname{Rm}(X,Y,Z,W)=\operatorname{Rm}(Z,W,X,Y).

It also obeys the first : the cyclic sum of R(X,Y)ZR(X,Y)Z over X,Y,ZX,Y,Z vanishes. These identities depend on both metric compatibility and vanishing torsion of the Levi–Civita connection; see Lee, Chapter 7, “Curvature”.

Geometric information

For a two-plane spanned by linearly independent u,vu,v, the sectional curvature is

K(u,v)=Rm⁡(u,v,v,u)g(u,u)g(v,v)−g(u,v)2.K(u,v)=\frac{\operatorname{Rm}(u,v,v,u)} {g(u,u)g(v,v)-g(u,v)^2}.

Contractions of Rm⁡\operatorname{Rm} produce the and scalar curvature. Thus the Riemann tensor contains all sectional curvatures, while its contractions retain progressively less directional information.

Parallel transport around a small loop differs from the identity to first order in the enclosed area by the curvature operator. This gives the tensor its interpretation as infinitesimal holonomy.

Examples and non-examples

with its standard metric has R=0R=0. The unit round sphere has constant positive sectional curvature, whereas hyperbolic space has constant negative sectional curvature.

A connection on an arbitrary also has , but that endomorphism-valued 22-form is called the Riemann curvature tensor only when the connection is the Levi–Civita connection on TMTM. A nonzero Christoffel symbol is not by itself curvature; such symbols can be nonzero in curvilinear coordinates on flat Euclidean space.

Conventions and scope
References
  1. John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Graduate Texts in Mathematics 176, Springer, 2018. Publisher record. Relevant: Chapter 7, “Curvature.”