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I know that it is well defined too say that a differential $k$-form on a manifold M with dimension n is somewhat regarded as a map from the tangent space of the manifold $M$ to $\mathbb{R }$ for each $p\in M,$ then we can define $w$ as the $k$-form where:

$$w=\sum_{i_1,...,i_k=1}^n w_{i_1,...,i_k}du^{i_1}\wedge...\wedge du^{i_k}$$

My question is, what is the geometric significance of the $k$-form defined as above when $w(p)\ne 0?$

Aurora Borealis
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Differential forms eat vector fields and spit out numbers (or in a more esoteric language: they live in the cotagent bundle).

The geometric significance of a top form $ω$ (i.e. an $n$-form in a $n$-dimensional manifold)is somewhat straight forward: Given $n$ vectors $u_1,...,u_n$ on $T_pM$,

$w(u_1,...,u_n)=c_p\cdot \det(u_1,...,u_n)$=

$c_p\cdot Vol(\text{the siplex having as edges the vectors $u_1,...,u_n$})$

$c_p$ is just a number which varies smoothly on the point $p$.

Now a $k$-form on an $n$ dimensional manifold (not necessarily a top form) does almost the above but before spitting out the volume it projects. So say for example that we have the form $ω(a,b,c)=3(a+b)dx\wedge dy$ on $\mathbb{R^3}$ and we want to compute what it does on $u_1,u_2 \in T_{(1,0,0)}\mathbb{R^3}$. $ω$ would take the 2 vectors, project them on the $x,y$ plane and compute the volume of the parallelogram spaned by these 2 projections. Finally it would multiply the volume by some factor, in our case $3(1+0)=3$.

By analogy, if we had $ω(p)=f(p)dxdy + g(p)dydz+h(p)dzdx$ on $\mathbb{R^3}$ then for any two vectors in the tangent space at the point $p$ it would add the scaled volumes of the parallelograms of the projections at the planes $xy$,$yz$ and $xz$ respectively.

To see why this happens you can check David Bachmann's really good introductory book "A geometric approach to Differential forms" it really is a very nice introduction to the subject with lots of excerises and examples.

Nick A.
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  • I am sorry I still dont understand what kind of action the $k$-form performs on just any vector $p\in M$, I perfectly understand what it does to the vectors in the tangent space, but the what it does to just the vectors on the manifold seems still very confusing. – Aurora Borealis Apr 12 '18 at 12:53
  • @AuroraBorealis I see the source of confusion. A differential form takes vectors and does somethign to them and then returns numbers. Where do these vectors live ? Not on the manifold, but on the $T_pM$ , the tangent space of each point on the manifold. It may happen that the differential form treats the vectors from one tanget space different than the vectors from another tangent space. So what it does to vectors depends on the poitn $p$. for example the for the differential form from the 3d paragraph : (cont)... – Nick A. Apr 12 '18 at 13:57
  • $ω{(1,0,0)}(u_1,u_2)=3(1+0)\cdot dx(u_1,u_2)\wedge dy(u_1,u_2)$ . On the other hand $ω{(-1,2,3)}(v_1,v_2)=3(-1+2)\cdot dx(v_1,v_2)\wedge dy(v_1,v_2)$. So a differential form is a *collection of alternating forms in each tangent space. Note that the $u_i$ live on $T_{(1,0,0)}\mathbb{R^3}$ while the $v_i$ live on $T_{(-1,2,3)}\mathbb{R^3}$. – Nick A. Apr 12 '18 at 14:04
  • This makes it alot more clear. Thank you very much. – Aurora Borealis Apr 12 '18 at 14:06
  • @Nick.A https://math.stackexchange.com/questions/2733365/existence-of-a-coordinate-system Could you help me explain this thread? Because I am still struggling to understand this and your explanations on this area seems easy for me to understand. I would appreciate it. – Aurora Borealis Apr 14 '18 at 05:02