Global choice of eigenvectors on an open surface

Question

Let $(M^2,g)$ be a noncompact orientable Riemannian surface without boundary. Let $A \in \Gamma(\operatorname{Sym}(TM))$ be a section of the bundle of symmetric endomorphisms of $TM$, that is, for each $p \in M$, the linear map $A_p : T_p M \to T_p M$ is symmetric with respect to the inner product $g_p$. Assume that the eigenvalues $\lambda_1(p), \lambda_2(p) \in \mathbb{R}$ of $A_p$ are distinct for all $p \in M$.

Do there exist two smooth globally defined unit vector fields $E_1, E_2$ on $M$ such that $E_i(p)$ is an eigenvector of $A_p$ associated with $\lambda_i(p)$ for $i=1,2$ and all $p \in M$?

Does the topology of $M$ impose any restrictions on the existence of such vector fields?

Answer 1

Not necessarily. To construct a counter-example, start from the other direction. Suppose that the tangent bundle of $M$ can be split as the direct sum $TM = L_1\oplus L_2$ where $L_1$ and $L_2$ are non-trivial smooth line bundles. Then you can easily construct a metric $g$ on $M$ such that $L_1$ and $L_2$ are $g$-orthogonal. Let $A:TM\to TM$ be the linear transformation that is the identity on $L_1$ and minus the identity on $L_2$. Then $A$ is symmetric with respect to $g$, but the two eigenbundles of $A$ have no non-trivial sections.

A simple example of such a splitting is to let the (orientable) cylinder $M$ be the quotient of the plane $\mathbb{R}^2$ by the translation $(x,y)\mapsto (x{+}\pi,y)$, and let $L_1$ be the bundle spanned by the vector field $Z = \cos x\,\partial_x + \sin x\,\partial_y$ (which is only defined up to a sign on $M$), while $L_2$ is spanned by the vector field $W = -\sin x\,\partial_x + \cos x\,\partial_y$ (again defined only up to a sign on $M$).