Intuition behind the Riemann curvature as a map $\bigwedge^2 T^*M\to \bigwedge^2 T^*M$?

Question

Given a (pseudo)-Riemannian manifold $(M^n ,g)$, one can naturally define the Levi-Cevita connection $\nabla_g: \Gamma(TM)\to \Omega^1(M,TM)$ as the unique metric-compatible, torsion-free connection on $TM$. This allows us to define the Riemann Tensor: $R: TM\otimes TM\to \mathrm{End}(TM)$ by $$R(\xi_1,\xi_2)(\xi_3):=([\nabla_{g,\xi_1}, \nabla_{g,\xi_2}]-\nabla_{[\xi_1,\xi_2]})\xi_3.$$ Since the map $(\xi_1,\xi_2,\xi_3,\xi_4)\mapsto \langle R(\xi_1,\xi_2)\xi_3,\xi_4\rangle$ is antisymmetric in the first and second slots, as well as in the third and fourth slots, it descends to a map $\bigwedge^2TM\to \bigwedge^2 T^*M$ or equivalently $\bigwedge^2 T^*M\to \bigwedge^2 T^*M$ by precomposition by the correct bundle isomorphism induced by $g$. Is there a nice way to think of this endomorphism, i.e. as some sort of shearing/rotation of oriented planes, or is it just a mathematical mirage?

Answer 1

From my personal point of view, the answer is NO. I would be happy if somebody shows I am wrong.

As you know, given a vector space $V$ with a non-degenerate bilinear form $g$ you can pass from $V$ to $V^*$ and from maps $\varphi: V \to V$ to maps $\varphi': V^*\to V$, $\tilde{\varphi}: V^*\to V^*$,... and so on, in a very straightforward way. The meaning of these "variations" is nothing but change the interpretation of elements of $V$ from "little arrows" to "families of hyperplanes slicing the total space" (that is, covectors). Depending of the situation you can be more comfortable with one description or with another.

Having said that, for me the better description for the Riemann curvature tensor is not in the way $$ T^*M\otimes T^*M \to T^*M\otimes T^*M. $$

On the contrary, I prefer $$ TM\otimes TM\otimes TM \to TM, $$ because it let you see the Riemann curvature tensor as a generalization of the Gaussian curvature like the holonomy per unit area, in the case of surfaces.
That is, in a surface you can compute the Gauss curvature by performing a loop around a point and measuring how an arbitrary vector has rotated (holonomy) when it is parallel transported around the loop. It is independent of the choice of the vector, since the angle of two parallel transported vectors remains constant. Then you shrink the loop around the point and the limit of the measured rotation, per unit area, is the Gauss curvature.

If we want to generalize this construction for $n$-dimensional manifolds we have the problem that we have "too much space". So in this case:

1. We have to specify in which "plane" are we going to "throw" our loops. This direction is specified by the first two slots of the Riemann's curvature tensor.

2. Neither the initial or the final parallel transported vector along the loops have to belong to the chosen plane. What we have is a linear map sending the original vector to the difference of the final vector with the initial vector. These are, respectively, the third slot of the tensor, and the resulting vector of the Riemann's curvature tensor.

As you can see in the picture, and following your notation, $\xi_1,\xi_2$ specify a 2-direction, $\xi_3$ is the vector being parallel transported. The vector named $R$ is the difference. This generalizes Gaussian curvature from the point of view of holonomy because if we have $n=2$ everything remains "flat" and the interesting data is the "rotation" of the parallel transported vector $\xi_3$, which can be expressed by a single scalar.

You can find this approach in the wonderful book of Tristan Needham, Visual Differential Geometry.

The other expressions of the Riemann curvature tensor, I think, mean nothing new, but dealing with vectors from dual point of views, which can be more operatives in some situations.