Could any one give me hint how to show that the zero section of any smooth vector bundle is smooth?
Zero section is a map $\xi:M\rightarrow E$ defined by $\xi(p)=0\qquad\forall p\in M.$
Could any one give me hint how to show that the zero section of any smooth vector bundle is smooth?
Zero section is a map $\xi:M\rightarrow E$ defined by $\xi(p)=0\qquad\forall p\in M.$
Smoothness can be checked locally on $M$ and locally $E$ is trivial.
Can you use these two facts to conclude what you want?
Say $E\to M$ is a smooth vector bundle, and let $p\in M$. Then there is a neighborhood $U\subseteq M$ of $p$ and a smooth local trivialization $\Phi:\pi^{-1}(U)\to U\times\mathbb R^k$ of $E$ over $U$. What can you say about $(\Phi\circ\zeta)(q)$ where $q\in U$? Can you conclude that $\Phi\circ\zeta$ is smooth? Now remember that $\Phi$ is a diffeomorphism.