sectional curvature determines Riemann curvature tensor

Theorem 1.

The sectional curvatureMathworldPlanetmath operator ΠK(Π) completely determines the Riemann curvature tensorMathworldPlanetmath.

In fact, a more general result is true. Recall the Riemann (1,3)-curvature tensor R:TMTMTMTM satisfies

(x,y,z,t)+(y,z,x,t)+(z,x,y,t) =0First Bianchi identity (1)
(x,y,z,t)+(y,x,z,t) =0 (2)
(x,y,z,t)-(z,t,x,y) =0, (3)

where (x,y,z,t):=g(R(x,y,z),t), and the sectional curvature is defined by

K(Π=span{x,y})=g(R(x,y,x),y)g(x,x)g(y,y)-g(x,y)2. (4)

Thus Theorem 1 is implied by

Theorem 2.

Let V be a real inner product spaceMathworldPlanetmath, with inner productMathworldPlanetmath -,-. Let R and R be linear maps V3V. Suppose R and R satisfies

  • Equations (1),(2),(3), and

  • K(σ)=K(σ) for all 2-planes σ, where K,K are defined by (4) using -,- in of g(-,-).

Then R=R.


(x,y,z,t) :=R(x,y,z),t
(x,y,z,t) :=R(x,y,z),t.
Proof of Theorem 2.

We need to prove, for all x,y,z,tV,


From K=K, we get (x,y,x,y)=(x,y,x,y) for all x,yV. The first step is to use polarization identityPlanetmathPlanetmath to change this quadratic formMathworldPlanetmath (in x) into its associated symmetric bilinear formMathworldPlanetmath. Expand (x+z,y,x+z,y)=(x+z,y,x+z,y) and use (3), we get


So (x,y,z,y)=(x,y,z,y) for all x,y,zV.

Unfortunately, the form (x,y,z,t) is not symmetricPlanetmathPlanetmath in y and t, so we need to work harder. Expand (x,y+t,z,y+t)=(x,y+t,z,y+t), we get


Now use (2) and (3), we get

(x,y,z,t)-(x,y,z,t) =(x,t,z,y)-(x,t,z,y)

So (x,y,z,t)-(x,y,z,t) is invariant under cyclic permutationMathworldPlanetmath of x,y,z. But the cyclic sum is zero by (1). So


as desired. ∎

Title sectional curvature determines Riemann curvature tensor
Canonical name SectionalCurvatureDeterminesRiemannCurvatureTensor
Date of creation 2013-03-22 15:55:09
Last modified on 2013-03-22 15:55:09
Owner kerwinhui (11200)
Last modified by kerwinhui (11200)
Numerical id 10
Author kerwinhui (11200)
Entry type Theorem
Classification msc 53B21
Classification msc 53B20