3
$\begingroup$

Would somebody be able to prove that the whitney sum of the normal and tangent bundles of a submanifold of R^n is trivial? Would apreciate a detailed proof...I'm struggling a little. Tina

1 Answers 1

3

Let $f:M^m\rightarrow \mathbb{R}^n$ be an embedding, let $\tau(M)$ be the tangent bundle of $M$, $\nu (M)$ the normal bundle of the embedding $f$. Then we have the formula

$\tau(M)\oplus\nu(M)\cong f^*\tau(\mathbb{R}^n)$

where $f^*\tau(\mathbb{R}^n)$ denotes the pullback of the tangent bundle of $\mathbb{R}^n$ along $f$ (which you could think of as the tangent bundle of $\mathbb{R}^n$ restricted to $f(M)$).

$\tau(\mathbb{R}^n)$ is canonically isomorphic to $\mathbb{R}^n \times\mathbb{R}^n$, and so its pullback is also trivial.

  • 0
    Why is the sum of the normal and tangent budles of $M$ isomorphic to the pullback bundle of the tangent bundle $\tau (R^n)$?2014-01-31