6

What properties make alternating forms so desirable in differential geometry?

hardmath
  • 37,015
Titiana.J.
  • 61
  • 1
  • 2
    It does encapsulate the orientation of whatever you're integrating over in a natural way. I don't know whether that's the only reason, but to me that's enough. – Arthur Apr 20 '17 at 13:24
  • Please fix the title. – mlc Apr 20 '17 at 13:43
  • 2
    Alternating forms encapsulate and generalize the following idea: The area of the square spanned by $(1,0)$ and $(0,1)$ is $\det{\begin{pmatrix}1 &0\0 & 1\end{pmatrix}}=+1,$ but if we take the vectors in the opposite order (first $(0,1)$ and second $(1,0)$) then we get a square of "area" $\det{\begin{pmatrix}0 & 1\1 & 0\end{pmatrix}}=-1.$ In particular, I think a correct answer should include the interpretation of the determinant as an oriented volume factor. This is intimately related to exterior algebra. I am not qualified to say more. – Will R Apr 20 '17 at 15:52
  • See more answers at https://math.stackexchange.com/questions/1160565/why-integrating-only-alternating-forms (of which this is basically a duplicate). – Toby Bartels Mar 22 '23 at 18:35
  • Does this answer your question? why integrating only alternating forms? – Toby Bartels Mar 22 '23 at 18:36

1 Answers1

3

It should be noted that not only alternating forms are important ingredients in differential geometry. Many other tensors have been used and studied along the years. Riemannian metrics and various forms of curvature are examples of that.

To the best of my knowledge, the property which distinguishes alternating tensors, that is, differential forms, is the fact that one can integrate them. As mentioned in the comments, everything starts with the determinant, which is alternating. Then, by the formula for changing variables in multiple integrals, it follows that the integral of a $k$-form on a $k$-dimensional oriented manifold is intrinsically well defined.

Amitai Yuval
  • 19,308