Let be a Riemannian manifold with metric , Levi-Civita connection , and curvature tensor . Define Apparently people sometimes refer to this as the curvature tensor, though it is probably not too confusing.

**Some algebra **

Recall also the following three identities, proved here:

- (Skew-symmetry)
- (Skew-symmetry)
- (Bianchi identity)

I claim now that there is a type of symmetry:

This is in fact a general algebraic lemma.

Lemma 1Let be a real vector space and let be a quadrilinear map satisfying the three bulleted identities. Then it satisfies the boxed one.

The proof is some slightly messy algebra, which I’ll only sketch. There is a geometrical way of thinking about this that Milnor presents in *Morse Theory*.

Let the Bianchi identity as written above be denoted by . If we add and use skew-symmetry several times, we obtain

Now using gives

and suitably interchanging all the variables gives the result.

**Sectional curvature **

Notation as above, for a 2-dimensional subspace , define the **sectional curvature** as

if form an orthonormal basis for .I now claim this is well-defined.

Proposition 2If span , then In particular, depends only on and is well-defined.

Indeed, write . Then this is a computation depending on the previous identities already proved. (Messy algebra and blogging do not mix well.) ** **

**Sectional curvature determines **

The sectional curvature actually encodes all the information contained in . Indeed, if we had two Riemannian metrics on the same manifold with curvature tensors with the same sectional curvature in all two-dimensional planes, then

for all .

Consider the difference . Then it satisfies all the four identities in the first section of this post, along with (by skew-symmetry again). Also by the fourth identity, is symmetric in , so the skew-symmetry just proved implies

In particular, we get another skew-symmetric identity:

Applying this again gives

and the Bianchi identity clearly gives that .

I should say something about the geometric interpretation about all this. If , then for a small neighborhood of is a surface in . Then the Gauss curvature of that surface is .