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I'm a bit confused by notation. The abstract tensor product space of a vector space $V$ with itself, $k$-times, is the free real vector space of their cartesian product where we identify certain $k$-tuples with zero. We write this as $V \otimes \cdots \otimes V$.

But in differential geometry, the space of contravariant $k$-tensors $T^k(V)$ is written the same way. Does this mean the usual abstract tensor product space (in the case where the factors are all the same) is the space of contravariant tensors on $V$?

For some reason I had the intuition that covariant tensors are the "default" and contravariant tensors are a bit weird, so this is somewhat surprising.

Brendan Langfield
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1 Answers1

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Yes, that is true.

In differential geometry, elements of the tangent space are contravariant vectors, and tensor products of them are contravariant tensors (with upper indices).

This means that the space of fully contravariant rank-$n$ tensors at a point is the $n$-fold tensor product of the tangent space. (On the other hand, the space of fully covariant tensors is a tensor product of the cotangent space).

I agree that this feels unintuitive, because intuitively tangent vectors feel more concrete than covectors, so they "should" not have the name that suggests they're the weird case. But the terminology is what it is.

On the other hand, when we move to higher-rank tensors, the covariant case is arguably the one that is more familiar. A rank-two covariant tensor can be thought of as something that eats two tangent vectors and spits out a scalar -- that is, just a bilinear form which we're familiar with from linear algebra. It is harder to form an intuition about what a rank-two contravariant tensor represents. Something that eats a gradient and produces a tangent? Or something that eats two gradients and produces a scalar? Neither of these really paints a particularly vivid picture in my mind.

Another sort-of argument is that we speak of (tangent) "vectors" and "covectors", using the common naming convention that a "co-something" is the weird, opposite, dual to a "something". However, if we want to form adjectives for how the different indices of a tensor transforms, it would sound confusing to call them "variant" versus "covariant", so we need to press a different prefix into service for parts that transform like a vector does -- hence "contravariant", because "contra-" is at least linguistically opposite to "co-".

(And it would have been even more confusing to swap the terms around and say that a "vector" transforms "COvariantly", whereas a "COvector" transforms "CONTRAvariantly").

Troposphere
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