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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 ?

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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.

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I think one reason is a lot of basic differential geometry constructions are anti-symmetric by nature. For example, no matter which connection you choose, the curvature is always defined by $$ K(X,Y)Z=\nabla_{X}\nabla_{Y}(Z)-\nabla_{Y}\nabla_{X}(Z)-\nabla_{[X,Y]}(Z) $$ and it is clear from the definition that $K(X,Y)(Z)=-K(Y,X)Z$. So the curvature is a so called two form. This and the requirement that connection has to be compatible with the metric give rise to lots of identities for the Riemannian curvature tensor. Since curvature is the heart of differential geometry, it is not surprising that the anti-symmetric properties has to play a crucial role.

The other reason has to do with de Rham cohomology. The anti-symmetric nature of the forms acted nicely when we introduce exterior differentiation. And a lot of constructions - volume forms, characteristic classes, etc are natural from this perspective. However, I think both symmetric and anti-symmetric qualties are important: the metric tensor is symmetric, the Christoffel symbols for a symmetric connection is symmetric, and they are equally important - you cannot define anything without the metric in geometry.

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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.

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