I believe I have a reasonable basic understanding of orientation. Yet, i'm finding myself utterly confused when facing a specific question. Here are several examples:
Exhibit an ordered basis of positive orientation for $S^2$ (as a boundary of $B^3$) at an arbitrary point point $p=(a,b,c)$.
Let $f: S^2 \to (-1,1)$, be gicen by $f:(x,y,z) \mapsto z$. Exhibit an ordered basis of positive orientation for a typical point on $f^{-1}(t)$.
Show that the boundary orientation of $S^k = \partial B^{k+1}$ is the same as its preimage orientation given by:
$$g:\mathbb{R}^{k+1} \to \mathbb{R}, g(x)= |x|^2$$
(The questions are taken from the book "differential topology" by Guillemin and Pollack).
My issues:
- I don't quite understand how an answer is supposed to look like. Couldn't i just say $\{v_p,w_p\}$ is a basis for $T_p S^2$ so (if $n_p$ is the outward pointing normal) according to whether $sign(n_p,v_p,w_p)$ is positive or negative i can switch $v_p$ and $w_p$ and obtain a positive orientation?
- Same problem with (1). Only now there's another issue. I'm not sure what's the correct formal way of computing an orientation of a preimage like $f^{-1}(t)$. Would it be right to look at $(T_p f o \pi_p)^{-1}(1)$ where $\pi_p: T_p S^2 \to T_p (f^{-1}(t))$ is the projection?
- Same as (2), plus some general unfocused non-specific confusion.
I must say that I didn't have any problems with the chapter itself, everything was done in a coordinate-free way that didn't raise these problems.
Thanks for the help.
EDIT: In the book an orinetation is defined as a continuous choice of equivalence class on the tangent space at each point where the relation is:
$$v \sim w \iff Av=w \text{ for some $A$ with } |A| > 0$$
I do feel comfortable with the definition of an atlas with positive determinant transition functions as well.
To get you started, here are some hints for question 1.
Start with you are given: a point $p \in S^1$ with coordinates $p = (a,b,c)$.
Now, it is fine to use the symbols $v_p,w_p$ as names representing positively oriented basis vectors for the tangent space of $S^2$ at the point $p$. But don't fool yourself: writing down names for those vectors is not the same as giving formulas for them. Your formulas should be functions of the coordinates $a,b,c$ of the point $p$. In other words, your answer should have the format
where $f(a,b,c)$ is some completely explicit formula, as are $g$, $h$, $k$, $l$, and $m$.
The key geometric fact you need about $S^2$ is that the normal vector of $S^2$ at the point $p=(a,b,c)$ is precisely $\langle a,b,c\rangle$. Then you also need that the tangent space at $p$ is precisely the orthogonal subspace of the normal vector.
So then, you just need to do some linear algebra to find formulas for the functions $f,g,h,k,l,m$ so that the matrix $$\begin{pmatrix} a & b & c \\ f(a,b,c) & g(a,b,c) & h(a,b,c) \\ k(a,b,c) & l(a,b,c) & m(a,b,c) \end{pmatrix} $$ has positive determinant, and so that its second and third rows are orthogonal to its first row. The key words here are "Gram-Schmidt orthogonlization".