Sphere orientation form

• Feb 7th 2011, 04:41 PM
Bongo
Sphere orientation form
Hello,

i have read in wikipedia, that a orientation form (volume form) of the sphere S^m is given by: $\displaystyle w=\sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \wedge...\wedge dx_{i-1} \wedge dx_{i+1} \wedge...\wedge dx_{n+1}$

I don't understand the notation. Ok w is a map from the sphere into the set of alternating tensors:
$\displaystyle w:S^m->\Lambda^m (S^m)$.

but what is $\displaystyle w(x_1,...,x_m+1)=$?
and what does dx_i mean?

Can you please explain it to me?

Regards
• Feb 9th 2011, 04:48 AM
xxp9
Giving examples may help you to understand. Let n=2 we have the standard sphere $\displaystyle S^2$ embedded in $\displaystyle R^3$:
$\displaystyle \{(x,y,z)|x^2+y^2+z^2=1\}$. Thus
$\displaystyle \omega=x dy\wedge dz+y dz \wedge dx + z dx \wedge dy$.
This is a top form defined on $\displaystyle S^2$ and you can easily check that it is nowhere zero. So it is a volume form thus defines an orientation of $\displaystyle S^2$.
Actually it is the volume element so that integrating it will get the surface area: $\displaystyle \oint_{S^2} \omega = \int_{D^3} d\omega = \int_{D^3} 3 dx\wedge dy \wedge dz= 3 Volume(D^3) = 4\pi$