# Vector Fields in a Riemannian Manifold

Suppose $(M,g)$ is a Riemannian manifold. Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator? Thanks

• Clearly Killing vector fields have this property. But in general, I guess that one can think of such vector fields as generators of "isospectralities" (instead of isometries). What would be a nice example of a manifold with such a vector field that is not Killing? Aug 21 '15 at 0:37
• Actually I'm not sure that my statement on "isospectralities" makes sense: the principal symbol of the Laplace operator is the metric itself. So doesn't this imply that every vector field that commutes with the Laplacian must be Killing? Aug 21 '15 at 3:48
• That is a good point but then should we consider the principal symbol of the commutator? That would be the poisson bracket of the two principal symbols
– Ali
Aug 21 '15 at 4:13

I give a geometric explanation of the calculations of Willie, which simultaneously elaborates the suggestion of Deane.

The flow of a vector field commuting with Laplacian preserves the Laplacian and therefore its symbol which is the inverse of the metric (i.e., $g^{ij}$). Then, the flow of the vector field preserves the metric and the vector field is Killing

A related question is

is $\nabla \cdot ( c^2 \nabla)$ a Laplace-Beltrami operator?

Acting on scalars we have \begin{align} [\triangle_g, \mathcal{L}_X] &= [g^{ab} \nabla_a \nabla_b ,\mathcal{L}_X] \\ &= [g^{ab} \nabla_a, \mathcal{L}_X]\nabla_b \\ &= {}^{(X,0)}\pi^{ab} \nabla_a \nabla_b + g^{ab} [\nabla_a, \mathcal{L}_X] \nabla_b \\ &= {}^{(X,0)}\pi^{ab} \nabla_a\nabla_b + g^{ab} {}^{(X,1)}\pi_{ab}^c \nabla_c \end{align} where the deformation tensors are $${}^{(X,0)}\pi_{ab} = \mathcal{L}_X g_{ab}$$ and $${}^{(X,1)}\pi_{ab}^c = \frac12 g^{cd} \left( \nabla_a {}^{(X,0)}\pi_{bd} + \nabla_b {}^{(X,0)}\pi_{ad} - \nabla_d {}^{(X,0)}\pi_{ab} \right)$$ from this you see that for the commutator to vanish identically it is necessary and sufficient that $X$ is Killing.
• @WillieWong Is there a Riemannian metric on $\mathbb{R}^2$ or the punctured plane such that the vander pol vector field would be a Killing vector field. In this case the space of Harmonic maps would be invariant under flow then one can think to compute the algebraic index of such operator (to have some dynamical information on X) Jun 5 '17 at 20:29