This question begins is related to this question on physics.SE Uniqueness of Riemann Curvature Tensor, which asks roughly "what tensors can we make locally out of just the metric tensor? We can clearly make a (3,1) tensor out of just the metric, but we can't seem to locally make a vector field out just the metric.
I gave an answer there on which I'd appreciate comments. What I'd like to ask here is
Can we formalize the intuition of making a tensor out just the metric tensor?
The formalization I have come to is: constructing a tensor of type $A$ out of just a tensor of type $B$ is the same as a natural transformation between the two functors that assign to each manifold the space of tensors of type $A$ & $B$ respectively.
When I say a tensor "of some type", I mean specifying the number of upper and lower indices as well as possibly antisymmetrization/symmetrization of indices and any other $GL(n)$ invariants you can slap on. So symmetric, nondegenerate (0,2) tensor is a type.
(1) Is the above definition good? Does it capture the intuitive idea of "making a tensor field out of just another one"? Or am I missing something simple? Does this have an established name (other than natural transformation)?
The transformations from a fixed type form a tensor algebra inherited from $T(V)$: we can add, multiply by constant scalars, tensor products, contractions, symmetrizations - any $GL(n)$ invariant operation. This is kind of trivial since it doesnt know anything about the manifold, it's inherited from the vector space. So we just want to talk about the generators of this algebra, which always includes at least the trivial objects (i) identity map and (ii) map everything to $1$.
(2) Have the generators of this algebra been classified, or studied?. Or is this just asking someone to "list all the geometrical invariants ever" using fancy words? Assuming it is, some bonus questions
(3)Can anyone find a counterexample to this extreme conjecture: The entire algebra is generated by (i) metric curvature object, (ii) exterior derivatives and (iii) Hodge duals all composed with tensor algebra? (As a physicist these are the only natural geometric objects I can think of. Sad physicist)
(4)Is the only nontrivial transformation of a 2-form in $d=4$ the exterior derivative?
(5)Is there any nontrivial transformation of a symmetrized tensor of type (0,3)?
Sorry if that's a lot of questions, I just find this pretty cool.
Caveats 'n such
- Everything is smooth. I really want real analytic but I will settle for $C^\infty$.
- Probably we need to work on connected manifolds to remove some issues involving "constant" scalars taking values on different components.
- The natural transformation have lots of other algebraic structure I dont know what to do with, including regular composition.
- Edited drastically from an earlier version to pointless wall of words. See the earlier version if you want more words.