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In differential geometry vectors and tensors can be defined without using transformation laws (thinking of vectors as operators acting on smooth functions from the manifold to $\mathbb{R}$). However I have never seen that approach on tensor densities. Is there any definition of tensor densities that doesn't involve behavior under certain transformation laws?

MRAA
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    Take a look at Explain densities to me please! – peek-a-boo Sep 15 '21 at 22:05
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    Thats a great question and I asked it myself some time ago. What I found is the following: Maybe you know that tensor fields are mathematically usually defined to be section of the tensor product bundle of tangent and cotangent spaces of a manifold. This is a purely geometrical definition without using coordinates (see for example Lee's book for a precise discussion). Now, tensor densities can be defined in a similar fashion globally, by taking the tensor product of the tensor bundle above with the so-called "density bundle". Sections of this construction are precisely tensor densitites. – G. Blaickner Sep 16 '21 at 08:18
  • Thank you peek-a-boo and G. Blaickner. I didn't knew of this! The question peek-a-boo suggested me to read explains precisely the density bundle. – MRAA Sep 16 '21 at 16:46

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