I will now review some differential geometry. Namely, I’ll recall what it means to have a connection in a complex vector bundle ${E}$, and construct its curvature as an ${E}$-valued global 2-form.

Now there is a fancy, clean approach to the theory of connections and curvature on principal bundles over a group (and a vector bundle basically corresponds to one such over $GL_n$). This approach is awesomely slick and highly polished: basically, it axiomatizes the intuitive idea that a connection is a way of identifying different fibers of a vector bundle (via parallel transport). So what is a connection on a principal bundle over a manifold? It’s a compatible system of defining whether tangent vectors are horizontal: the horizontal curves are those that correspond to a parallel transport. Then all the comparatively ugly index-filled results in the classical approaches get transformed into elegant, short results about Lie-algebra valued differential forms.

In fact, the whole Chern-Weil business can be developed using this formalism, and it becomes very slick. But I would like to do it in a slightly less fancy way, using the Cartan formalism: this essentially amounts to working in frames systematically. Here a frame is a local system of sections which is a basis for a vector bundle, and constitutes a generalized form of local coordinates. We can formulate the notions of connections and curvature in terms of frames (they’re systems of forms associated to each frame that transform in a certain way).

The theory has a super-optimal amount of index-pushing to it, but nonetheless, it is one I would like to gain comfort with, e.g. because Griffiths-Harris use it in their book. When one wants to actually prove concrete, specific results about certain types of manifolds (e.g. Kahler manifolds), it may be helpful to use local coordinates. An analogy: the theory of derived categories replaces the Grothendieck spectral sequence with the statement that the derived functor of the composite is the composite of the derived functors. But for concrete instances, the spectral sequence is still huge. (more…)

Let ${M}$ be a Riemannian manifold with metric ${g}$, Levi-Civita connection ${\nabla}$, and curvature tensor ${X,Y,Z \rightarrow R(X,Y)Z}$. Define $\displaystyle R(X,Y,Z,W) := g\left( R(X,Y)Z, W\right).$ 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) ${R(X,Y,Z,W) = -R(Y,X,Z,W)}$
• (Skew-symmetry) ${R(X,Y,Z,W) = -R(X,Y,W,Z)}$
• (Bianchi identity) ${R(X,Y,Z,W) + R(Z,X , Y, W) + R(Y, Z, X, W) = 0}$

I claim now that there is a type of symmetry: $\displaystyle \boxed{ R(X,Y,Z,W) = R(Z,W,X,Y).}$ This is in fact a general algebraic lemma.

Lemma 1 Let ${V}$ be a real vector space and let ${R:V \times V \times V \times V \rightarrow \mathbb{R}}$ 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 ${Rel_{X,Y,Z,W}}$. If we add ${Rel_{X,Y,Z,W}, Rel_{W,Y,Z,X}, Rel_{X,W,Z,Y}, Rel_{X,Y,W,Z}}$ and use skew-symmetry several times, we obtain $\displaystyle R(W,Y,Z,X) + R(Z,W,Y,X) + R(X,W,Z,Y) = 0.$

Now using ${Rel_{W,Y,Z,X}}$ gives $\displaystyle R(X,W,Z,Y) = R(Y,Z,W,X)$

and suitably interchanging all the variables gives the result.

Sectional curvature

Notation as above, for a 2-dimensional subspace ${S \subset T_p(M)}$, define the sectional curvature ${K(S)}$ as $\displaystyle K(S) := R( E, F, F, E)$

if ${(E,F)}$ form an orthonormal basis for ${S}$.I now claim this is well-defined.

Proposition 2 If ${X,Y \in T_p(M)}$ span ${S}$, then $\displaystyle K(S) = \frac{ R(X,Y,Y,X) }{(X,X)(Y,Y) - (X,Y)^2}.$ In particular, ${K(S)}$ depends only on ${S}$ and is well-defined.

Indeed, write ${X = x_1 E + x_2 F, Y = y_1 E + y_2 F}$. Then this is a computation depending on the previous identities already proved. (Messy algebra and blogging do not mix well.)

Sectional curvature determines ${R}$

The sectional curvature actually encodes all the information contained in ${R}$. Indeed, if we had two Riemannian metrics on the same manifold with curvature tensors ${R,R'}$ with the same sectional curvature in all two-dimensional planes, then $\displaystyle R'(X,Y,Y,X) = R(X,Y,Y,X)$ for all ${X,Y \in T_p(M)}$.

Consider the difference ${R' - R}$. Then it satisfies all the four identities in the first section of this post, along with ${B(X,Y,X,Y) = 0}$ (by skew-symmetry again). Also by the fourth identity, ${B(X,Y,X,Y')}$ is symmetric in ${Y,Y'}$, so the skew-symmetry just proved implies $\displaystyle B(X,Y,X,Y') \equiv 0.$ In particular, we get another skew-symmetric identity: $\displaystyle R(X,Y,Z,W) = -R(Z,Y,X,W)=R(Y,Z,X,W) .$

Applying this again gives $\displaystyle R(X,Y,Z,W) = R(Z,X,Y,W) ,$ and the Bianchi identity clearly gives that ${B \equiv 0}$.

I should say something about the geometric interpretation about all this.  If $S \subset T_p(M)$, then $\exp_p(S \cap U)$ for $U$ a small neighborhood of $0 \in T_p(M)$ is a surface in $M$.  Then the Gauss curvature of that surface is $K(S)$.

It turns out that the curvature tensor associated to the connection from a Riemannian pseudo-metric ${g}$ has to satisfy certain conditions.  (As usual, we denote by $\nabla$ the Levi-Civita connection associated to $g$, and we assume the ground manifold is smooth.)

First of all, we have skew-symmetry $\displaystyle R(X,Y)Z = -R(Y,X)Z.$

This is immediate from the definition.

Next, we have another variant of skew-symmetry:

Proposition 1 $\displaystyle g( R(X,Y) Z, W) = -g( R(X,Y) W, Z)$  (more…)