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We are working through old qualifying exams to study. There were two questions concerning normal bundles that have stumped us:

$1$. Let $f:\mathbb{R}^{n+1}\longrightarrow \mathbb{R}$ be smooth and have $0$ as a regular value. Let $M=f^{-1}(0)$.

(a) Show that $M$ has a non-vanishing normal field.

(b) Show that $M\times S^1$ is parallelizable.

$2$. Let $M$ be a submanifold of $N$, both without boundary. If the normal bundle of $M$ in $N$ is orientable and $M$ is nullhomotopic in $N$, show that $M$ is orientable.

More elementary answers are sought. But, any kind of help would be appreciated. Thanks.

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As it stands these are 'homework-type' problems... please shows us your attempts and where you are stuck. – Chris Gerig Jul 27 '12 at 3:27
up vote 3 down vote accepted

Some hints:

  1. (a) consider $\nabla f$. $\quad$(b)Show that $TM\oplus \epsilon^1\cong T\mathbb{R}^{n+1}|_M$ is a trivial bundle, then analyze $T(M\times S^1)$.

  2. Try to use homotopy to construct an orientation of $TN|_M$. Let $F:M\times[0,1]\rightarrow N$ be a smooth homotopy map s.t. $F_0$ is embedding, $F_1$ is mapping to a point $p$, then pull back (my method is parallel transportation) the orientation of $T_p N$ to $TN|_M$, and then use $TN|_M\cong TM\oplus T^\perp M$.

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