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Let $M^k \subset \mathbb{R}^{n+1}$, $M$ compact and $2k\leq n$. Show tha exist a $n$-hyperplane $H^n\subset \mathbb{R}^{n+1}$ such that if $\pi:H^{n}\oplus (H^n)^{\perp}\rightarrow H^{n}$ is the projection then $\pi\vert_{M}:M\rightarrow H^{n}$ is an immersion.
This is a exercise where suppose to use Sard's Theorem, let me a hint please.

gaoxinge
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Donyarley
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1 Answers1

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HINT: Can you use Sard's Theorem to show that there is a unit vector in $\Bbb R^{n+1}$ that belongs to no tangent space $T_xM$?

Ted Shifrin
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  • But, the tnagent space $T_x M$ is a linear subspace with dim$T_x M$=n emmbeded in $\mathbb{R}^{n+1}$ then theres at least one vector there that dont belongs to $T_x M$. I still do not see. – Donyarley Oct 13 '14 at 19:43
  • No, consider the map $f\colon TM \to\Bbb R^{n+1}$ given by $f(p,v) = v$. What do you know about its image, since $\dim TM = 2k<n+1$? – Ted Shifrin Oct 13 '14 at 21:26
  • It should not cover whole $\mathbb{R}^{n+1}$ – Donyarley Oct 16 '14 at 03:17
  • Or using Compact Whitney embedding – Donyarley Oct 16 '14 at 04:57
  • This is really part of the proof of Whitney Embedding. But, in fact, you don't need compactness to do this problem (or, in fact, to prove the strong version of Whitney Embedding). Sard's Theorem doesn't need compactness, just keeping track of dimensions. – Ted Shifrin Oct 16 '14 at 16:05
  • Since the compactness for $M$ the proof need be a particular case. A hint that I read say the next: Use the projectivization of the tangent bundle of M and define a function on $\mathbb{R}$P$^{n-1}$ – Donyarley Oct 16 '14 at 19:58
  • Way too complicated. Just show that if $H$ has normal vector $a$, with $a\notin T_xM$ for any $x\in M$, then $\pi|_M$ is an immersion. – Ted Shifrin Oct 16 '14 at 21:30