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EDIT: According to some comments on this post I revise the title to remove the misunderestanding.

Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated to the metric is denoted by $\Delta$.

Are there $n$ vector fields $X_{1},X_{2}, \ldots, X_{n}$ such that $\Delta=\sum \partial^{2}/\partial X _{i}^{2}$?

 

Are there some local obstructions?(However our question search for global vector fields $X_{i}s$)

The obvious motivation for this question is the usual metric on $\mathbb{R}^{n}$

EDIT: According to some comments on this post I revise the title to remove the misunderestanding.

Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated to the metric is denoted by $\Delta$.

Are there $n$ vector fields $X_{1},X_{2}, \ldots, X_{n}$ such that $\Delta=\sum \partial^{2}/\partial X _{i}^{2}$?

Are there some local obstructions?(However our question search for global vector fields $X_{i}s$)

The obvious motivation for this question is the usual metric on $\mathbb{R}^{n}$

Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated to the metric is denoted by $\Delta$.

Are there $n$ vector fields $X_{1},X_{2}, \ldots, X_{n}$ such that $\Delta=\sum \partial^{2}/\partial X _{i}^{2}$?

Are there some local obstructions?(However our question search for global vector fields $X_{i}s$)

The obvious motivation for this question is the usual metric on $\mathbb{R}^{n}$

EDIT: According to some comments on this post I revise the title to remove the misunderestanding.

Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated to the metric is denoted by $\Delta$.

Are there $n$ vector fields $X_{1},X_{2}, \ldots, X_{n}$ such that $\Delta=\sum \partial^{2}/\partial X _{i}^{2}$?

Are there some local obstructions?(However our question search for global vector fields $X_{i}s$)

The obvious motivation for this question is the usual metric on $\mathbb{R}^{n}$