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i am struggling to show that for a $n-dim.$ smooth submaifold $M \subset \Bbb R^k$ the normal bundle $NM=\dot \cup_{p\in M}(T_pM)^\bot \subset ι^*T \Bbb R^k\cong M\times \Bbb R^k $ is a smooth rank $(k-n)-dim.$ vektor bundle over $M$.

Where $ι: M \to \Bbb R^k $ denote the inclusion map.

help would be appreciated.

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  • $\begingroup$ Which part do you need help with? The rank? Do you know what the rank of the tangent bundle is? $\endgroup$ – ziggurism Nov 26 '17 at 1:17
  • $\begingroup$ should´t it be n . $\endgroup$ – Tsunà Nov 26 '17 at 1:19
  • $\begingroup$ Yes. The tangent bundle of an $n$-dimensional manifold is rank $n$. Therefore its orthogonal complement is rank $k-n$. $\endgroup$ – ziggurism Nov 26 '17 at 1:19
  • $\begingroup$ i thought it should be k-n because m is a submanifold in $\Bbb R^k$ $\endgroup$ – Tsunà Nov 26 '17 at 1:22
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    $\begingroup$ OK well I suppose you are already given those data for second property for the tangent bundle $TM$ and for the trivial bundle $M\times \mathbb{R}^k$. That is to say, for each $U$, you have $U\times \mathbb{R}^n$ and $U\times \mathbb{R}^k.$ How do you get $U \times \mathbb{R}^{k-n}$ from these? $\endgroup$ – ziggurism Nov 26 '17 at 1:59

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