Constructing a chart with coord. basis equal to given basis at one pt.

1. May 19, 2014

center o bass

Suppose we have a manifold $M$ and at $p \in M$ we have a basis for the tangent space of vectors $X_i$. Since $M$ is a manifold, there exists a local chart $(U,\phi)$ about $p$. Now the question is, given such a chart , how can we construct a new chart in a such that $X_i = \left. \frac{\partial}{\partial x^\mu} \right|_{p}$.

I know there is a theorem that says given commuting vector fields we can find a chart such that these vector fields locally is the coordinate basis of that chart.

However, I want to prove in a transparent manner the less general statement above; that we can construct a coordinate system such that the vector $X_i$ _in the tangent space at p_ is the coordinate basis $\left. \tfrac{\partial}{\partial x^\mu} \right|_{p}$ just at p.

2. May 19, 2014

Ben Niehoff

Use the standard construction for Riemann normal coordinates, but forget the fact that the basis is orthonormal.

3. May 19, 2014

pasmith

Let $(U,\phi)$ be a chart on $M$ whose domain contains $p$, and consider the chart $(U, A \circ \phi)$ where $A : \mathbb{R}^n \to \mathbb{R}^n$ is linear and invertible. This chart is smoothly compatible with the original chart $(U,\phi)$.

The transition function $A$ then pushes forward to a linear map $A_{*} : T_pM \to T_pM$, giving $$\left.\frac{\partial}{\partial x^i}\right|_p = \frac{\partial \tilde x^j}{\partial x^i} \left.\frac{\partial}{\partial \tilde x^j}\right|_p = (A_{*})_i{}^j \left.\frac{\partial}{\partial \tilde x^j}\right|_p$$ where $(x^i)$ are the coordinate functions of the chart $(U,\phi)$ and $(\tilde x^j)$ are those of the chart $(U, A \circ \phi)$. Since $\tilde x^j = A^j{}_i x^i$ we have that $$\frac{\partial \tilde x^j}{\partial x^i} = (A_{*})_i{}^j = A^j{}_i.$$ If ${X_j}$ is a basis for $T_pM$ we may then set $$\left.\frac{\partial}{\partial \tilde x^j}\right|_p = X_j$$ in the above to obtain $$\left.\frac{\partial}{\partial x^i}\right|_p = A^j{}_i X_j.$$

Last edited: May 19, 2014