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
Sum of normal bundle and tangent bundle.
3
$\begingroup$
manifolds
vector-bundles
1 Answers
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.
-
0Why 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