Why are differential forms more important than symmetric tensors?

Question

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What properties of differential forms makes them so useful in geometry ? And are there places in geometry where completely symmetric tensors are important objects of study ?

Answer 1

It is the natural language to describe the notions of volume and orientation. As you know from linear algebra, the determinant of an ordered list of $n$ vectors in $\mathbf R^n$ is a natural measure of the signed volume of the parallelepiped which they span. The determinant is naturally alternating, and can be described very simply using alternating forms. If $T$ is an endomorphism of a vector space $V$ of dimension $n$, then it induces an endomorphism $\Lambda^nT$ on the top exterior power $\Lambda^nV$, which is a one-dimensional vector space. An endomorphism of a one-dimensional space is just multiplication by a constant, and this constant is precisely $\det T$ (you could even take this as a definition).

This expresses the fact that the determinant is the "dilation factor" of $T$ acting on an infinitesimal volume element.

Symmetric tensors have their own uses, but they do not have the right properties to serve as a foundation for calculus.

Answer 2

Caveat: this is a very incomplete answer. Using the famous motto "A tensor is what transforms like a tensor", I would say that differential forms are fundamental because of their "link" with integration, and subsequent applications.

Just a little remark: working in characteristic 0, I would say that symmetric algebras are quite important: they play an major role, for example, in the Koszul duality theory or studying Lie algebras.