What is the importance and intuition behind the the contraction operator on tensors (or the trace of a matrix, for that matter)?


In addition, I see that one of the requirements for a covariant derivative (in the context of connections) is to commute with contraction. Why is that a natural requirement (like it would be for, say, the product rule)?



Tensor contraction and Covariant Derivative
I like the question. Below is a somewhat sketchy version of how I see this.


I think the importance of tensors and contraction of tensors originates from trying to do basic differential geometry or vector calculus from a co-ordinate-free point of view. The most basic objects are curves and velocity vectors of curves. The key observation is that the set of all possible velocity vectors at a single point in space is naturally an abstract vector space. Given any set of co-ordinates, the $n$ velocity vectors you get by holding all but one co-ordinates fixed form a basis of this vector space. If you change co-ordinates, then chain rule leads to the appropriate change of basis formula for velocity vectors. This space of all velocity vectors at a point in space is also known as the tangent space at that point. The construction described above defines the tangent space at each point in a smooth manifold.


But once you have a naturally defined abstract vector space, all of linear and multilinear algebra becomes available for use. For example, the dual space to the space of velocity vectors is the cotangent space, and at each point there is a natural contraction between a tangent vector and a cotangent vector. It is also natural to consider tensor products of copies of the tangent and cotangent spaces, leading to higher order tensors. Whether this is useful or not is not immediately obvious and becomes clear only after you go on further and discover how to construct tensors that have important geometric, physical, or topological meaning. But the upshot is that I like to think of a smooth manifold as being, among other things, a parameterized family of abstract vector spaces. Of course, that *is* exactly what a vector bundle is.


As for the covariant derivative, this can be introduced on its own. It arises, because if you try to find a natural co-ordinate-free way to differentiate vector fields, the best you can do is the Lie derivative or Lie bracket of two vector fields and a closer study of this shows that it is limited in its properties and usefulness. In particular, you want to, say, be able to define the directional derivative of one vector field $V$ in the direction of another vector field $W$. Moreover, just as the directional derivative at a point of a function in the direction $W$ depends only on the value of $W$ at that point (and not a neighborhood), you want the same property for the directional derivative of $V$. Moreover, the Leibniz rule for a scalar multiple of a vector field is a natural property to want. But you find that these properties don't determine a unique connection, so a connection is an additional geometric assumption. This is somehow unsatisfying.


The covariant derivative becomes more compelling, when it is introduced in the context of a Riemannian metric, which given the discussion about tangent vectors above, is a natural extension of the concept of Euclidean space and the inner product. Note that the Riemannian metric, as defined, is naturally a tensor. Next, there is something that, as far as I know is simply a miracle, namely that there is a unique naturally compatible torsion-free connection known as the Levi-Civita connection. After this, the value of tensors and the Levi-Civita connection is justified both because they correspond to naturally defined and interesting geometric concepts for submanifolds of Euclidean space and because they lead to so much insight and deep results in geometry, analysis, and topology.


