Let $0$ be a regular value of the smooth function $g:\mathbb{R}^n\rightarrow\mathbb{R}^p$ and $M=g^{-1}(0)$. Define the preimage orientaton of $M$ as such that $$[\triangledown g_1,\ldots,\triangledown g_p]\oplus[M]=[\mathbb{R}^n].$$

I've been asked for the following:

  1. Compare the preimage orientation of $\mathbb{S}^{n-1}=f^{-1}(0)$ ($f:\mathbb{R}^n\rightarrow\mathbb{R}$ given by $f(x)=\Vert x\Vert ^2 -1$) with the induced orientation on boundary of the closed unit ball, $\mathbb{S}^{n-1}=\partial B^n$ ($B^n$ with induced orientation of $\mathbb{R}^n$).
  2. Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a smooth function and $F:\mathbb{R}^3\rightarrow\mathbb{R}$ defined by $F(x,y,z)=z-f(x,y)$. Find positive basis for the tangent spaces to the graph of $f$, viewed with the preimage orientation given by $F$.

I don't know how to even start, so any detailed solution will be helpful to understand.

  • 1
    $\begingroup$ Can you precise what are your convention for the orientation of the boundary ? $\endgroup$ – user171326 Jun 14 '17 at 14:15
  • $\begingroup$ Do you know what the induced orientation on the boundary of the closed unit ball is? Or what the orientation of $B^n$ induced from $\mathbb{R}^n$? For 2, do you know how to calculate the gradient of $F$, and do you see how the graph of $f$ is equal to $F^{-1}(0)$, allowing you to apply the definition of preimage orientation to it? $\endgroup$ – Chill2Macht Jun 14 '17 at 14:30
  • $\begingroup$ @N.H. as defined by Spivak's "Calculus on manifolds" or by Munkres "Analysis on Manifolds" $\endgroup$ – Juan C. Cala Jun 14 '17 at 14:32

Here's a starting point: to compare these orientations, you only need to compare them at a single point. [Important question: Do you see why this is true?]

Let's look at the first problem, with $n = 3$. Then at the point $(1, 0, 0)$, the gradient of $f$ is something like $(2, 0, 0)$, right? And you need a basis for the tangent space with the property that $$ [(2,0,0), b_1, b_2] $$ is the positive orientation for $3$-space. I'm going to say that $e_2, e_3$ looks like a pretty good choice. (After all, the first vector is just $2 e_1$!)

Now you need to use the OTHER technique for finding a basis (the Spivak/Munkres one) at $(1,0,0)$, and see whether they agree or disagree.

And then you can try it again for $n = 2,$ and $n = 4$, and then you should be able to generalize pretty easily.


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