Is $TM\oplus NM \cong M\times \mathbb{R}^n $ valid if the manifold $M$ is immersed in $\mathbb{R^n}$?

Question

• I understand that if a manifold $M$ is embedded in $\mathbb{R}^n$, $TM\oplus NM \cong M\times \mathbb{R}^n $. Is this equation still valid if $M$ is immersed in $\mathbb{R}^n$?

• For example, if we let the Klein bottle $K$ immersed in $\mathbb{R}^3$, then the normal bundle $NK$ has fiber $\mathbb{R}$. Is $TK\oplus NK\cong K\times \mathbb{R}^3 $ valid?

• About the normal bundle, can see this question: Normal bundle of a non-orientable manifold

Answer 1

If $f : M \to M'$ is an immersion, then there is a short exact sequence of vector bundles on $M$:

$$0 \to TM \to f^*TM' \to \nu \to 0.$$

Here $\nu$ is the normal bundle which is defined to be the quotient bundle $\nu := f^*TM'/TM$ . I would avoid using the notation $NM$ for the normal bundle as it doesn't just depend on $M$, but rather the immersion $f$.

Choosing a bundle metric on $f^*TM$, we obtain an isomorphism $f^*TM' \cong TM\oplus\nu$. In particular, if $f : M \to \mathbb{R}^n$ is an immersion, then $f^*T\mathbb{R}^n \cong M\times\mathbb{R}^n$ as $T\mathbb{R}^n$ is trivial, and hence $M\times\mathbb{R}^n\cong TM\oplus\nu$. In the case of an immersion $K \to \mathbb{R}^3$, we see that we have an isomorphism of vector bundles $K\times\mathbb{R}^3\cong TK\oplus\nu$ where $\nu$ has rank one.