In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics that measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity. The curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.
Definition Let ( M , g ) {\displaystyle (M,g)} be a Riemannian or pseudo-Riemannian manifold, and X ( M ) {\displaystyle {\mathfrak {X}}(M)} be the space of all vector fields on M {\displaystyle M} . We define the Riemann curvature tensor as a map X ( M ) × X ( M ) × X ( M ) → X ( M ) {\displaystyle {\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\rightarrow {\mathfrak {X}}(M)} by the following formula where ∇ {\displaystyle \nabla } is the Levi-Civita connection:
R ( X , Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X , Y ] Z {\displaystyle R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z}
or equivalently
R ( X , Y ) = [ ∇ X , ∇ Y ] − ∇ [ X , Y ] {\displaystyle R(X,Y)=[\nabla _{X},\nabla _{Y}]-\nabla _{[X,Y]}}
where [ X , Y ] {\displaystyle [X,Y]} is the Lie bracket of vector fields and [ ∇ X , ∇ Y ] {\displaystyle [\nabla _{X},\nabla _{Y}]} is a commutator of differential operators. It turns out that the right-hand side actually only depends on the value of the vector fields X , Y , Z {\displaystyle X,Y,Z} at a given point, which is notable since the covariant derivative of a vector field also depends on the field values in a neighborhood of the point. Hence, R {\displaystyle R} is a ( 1 , 3 ) {\displaystyle (1,3)} -tensor field. For fixed X , Y {\displaystyle X,Y} , the linear transformation Z ↦ R ( X , Y ) Z {\displaystyle Z\mapsto R(X,Y)Z} is also called the curvature transformation or endomorphism. Occasionally, the curvature tensor is defined with the opposite sign. The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). Since the Levi-Civita connection is torsion-free, its curvature can also be expressed in terms of the second covariant derivative
∇ X , Y 2 Z = ∇ X ∇ Y Z − ∇ ∇ X Y Z {\textstyle \nabla _{X,Y}^{2}Z=\nabla _{X}\nabla _{Y}Z-\nabla _{\nabla _{X}Y}Z}
which depends only on the values of X , Y {\displaystyle X,Y} at a point. The curvature can then be written as
R ( X , Y ) = ∇ X , Y 2 − ∇ Y , X 2 {\displaystyle R(X,Y)=\nabla _{X,Y}^{2}-\nabla _{Y,X}^{2}}
Thus, the curvature tensor measures the noncommutativity of the second covariant derivative. In abstract index notation, R d
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