Is there a geometric intepretation of the trace of tensor on a Riemannian manifold?

Question

For a long time I thought the trace for matrices were just an elementary function with nice properties. But it is much more than that and really should be think of as a geometrical object (see Geometric interpretation of trace). For example considering the trace of a square matrix $M$ associated to a application $\phi_M \colon E\rightarrow F$ with $\dim E= \dim F = n$ doesn't make sense if $E\neq F$ and one should use the Trace only for endomorphism.

Now we consider tensors on a Riemannian manifold. There is the notion of "partial trace" : for a tensor $ T^{j_1\cdots j_n}_{k_1,\cdots k_m}$ we write

$$"\text{Trace}(T)"=T^{j_1\cdots j_{n-1}}_{k_1,k_{m-1}} :=\sum_i T^{j_1\cdots j_{n-1},i}_{k_1,k_{m-1},i}$$ when the trace has been made on the $n$ and $m$ coordinates. But now, as before, I understand the definition and I can use it to do computations but I just have no insight why this is a interesting object and what is the geometry behind it. How do I know I am not writting something that just doesn't make sense ?

An example of particular interest (and also fundamental in physics for general relativity) is the Riemann curvature tensor / Ricci tensor / scalar curvature.

$$ R^i_{jkl}\rightarrow R_{jl}\rightarrow R$$

My geometric intepretation of the Rieman curvature tensor is with parallele transport on small loops. I also know about the scalar curvature and the volume of the sphere. But I really can't see why one should be the trace of the other or in physics why the massive particles (which have no idea about tensors) should use the trace for the law of gravitation...

Answer 1

Quite generally, you implement the partial trace operation if you wish to discard local information in a coordinate-invariant way. On a manifold, the Ricci tensor gives you the average value of the sectional curvature along a particular direction, and its trace discards the directional information by averaging the curvature over the different directions.

A different context, which you may find instructive, appears in quantum physics, where the state of a system is described by a positive-definite matrix called the density matrix $\rho$. If the system is composed of two subsystems $A$ and $B$ and you wish to discard information about subsystem $B$ (for example, because it is far removed and you do not have access to it), then the partial trace $\rho_A=\sum_{i\in B}\rho_{i,i}$ describes the state of the system $A$ by itself.