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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:

  1. Exhibit an ordered basis of positive orientation for $S^2$ (as a boundary of $B^3$) at an arbitrary point point $p=(a,b,c)$.

  2. 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 $f^{-1}(t)$.

  3. 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:

  1. 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?
  2. 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?
  3. 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.

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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

  • $v_p = \langle f(a,b,c), g(a,b,c), h(a,b,c) \rangle$
  • $w_p = \langle k(a,b,c), l(a,b,c), m(a,b,c) \rangle$

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".

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  • $\begingroup$ Thanks, things do make a bit more sense now. $\endgroup$ – Saal Hardali Jul 20 '15 at 15:08

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