What properties make alternating forms so desirable in differential geometry?
2026-04-08 02:28:00.1775615280
Why are differential forms built up on alternating forms?
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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.