Prove that if (M, U) is a smooth n-dimensional manifold and p ∈ M, then there is a chart x: U → Rn such that x(p) = 0.

Question

This is a textbook problem from the book Introduction to Manifolds by Tu(Pg-41). I have no idea how to go about this problem. Any hints/suggestions would be greatly appreciated.

Answer 1

There is some chart $x:U\to R^n$. It maps $p$ into some point $x(p) \in R^n$, not necessarily $0$. But if you compose this chart with a smooth function $\phi: R^n \to R^n$, their composition $\phi \circ x:U\to R^n$ will be a chart compatible with the original chart (prove this using the definition of "compatible"), and it will therefore be part of the manifold's atlas (why? use the definition of smooth manifold). Now look for a very simple and certainly smooth function $\phi: R^n \to R^n$ that will make $\phi \circ x$ into what you want.