0
$\begingroup$

As we all know using multilinear map to define some concepts is very important because of its coordinate independent property,there are some questions I don't know how to see its inside motivation. 1.Why we want differential form to be totally antisymmetric? As for one reason,volumn form is unchanged under coordinate change if it has orientation.Then what about the differential form with order smaller than the dimension of its underlying space?What's the benefit to introduce antisymmtric in this case? Can someone give me an example to explain this. 2.exterior differentiate can be defined as coordinate dependent,but I learn from some book that intrinsic aspect of it is the connection on the global manifold?can anyone use some example to illustrate this compatibility?

If there is some misunderstanding of concepts in my question,you can point out. Appreciate in advance.

1 Answers 1

0
  1. The antisymmetry of differential forms is a direct consequence of their multi-linearity together with the fact we want the volumes of degenerate parallelepiped to be zero.

So,

$0=\alpha(v+w,v+w)=\alpha(v,v) + \alpha(v,w)+\alpha(w,v)+\alpha(w,w)=\alpha(v,w)+\alpha(w,v)$

  1. the exterior derivative is not defined w.r.t any connection on the manifold, it is part of the information which is implicitly contained inside the smooth structure of the manifold.