## “Noncommutative heat equation” — a strange generalization of Killing vectors for a flat metric

Let $(M,g)$ be a smooth (pseudo)Riemannian manifold with a flat metric $g$, and $X$, $Y$ be vector fields on $M$ such that $$L_X^2 (g)=L_Y(g). \hspace{70mm} \mbox{(1)}$$ where $L_Z$ is the Lie derivative along $Z$ and $L_X^2(g)\equiv L_X(L_X(g))$.

If $X=0$ then $Y$ is just a Killing vector for $g$ but

is there any (geometric?) interpretation for $X$ and $Y$ in the general case?

In fact I wonder whether Eq.(1), be it for fixed $g$ and unknown $X,Y$ or conversely for fixed $X,Y$ and unknown $g$, was studied systematically at all: I failed to find any relevant references.

Motivation: Equation (1) follows from the last formula in Proposition 5 from a paper on classification of compatible Hamiltonian structures that I have recently come across. It looks a bit like the heat equation for the metric, hence the title.

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Suppose $M$ is compact and $g$ is Riemannian. It is claimed that if $g$ has non-positive Ricci curvature, then either the Ricci curvature of $g$ is somewhere negative and both $X$ and $Y$ are $0$, or $g$ is Ricci flat and both $X$ and $Y$ are parallel. To begin with, do not suppose anything about the Ricci curvature of $g$. Let $D$ be its Levi-Civita connection and raise and lower indices using $g_{ij}$ and the inverse symmetric bivector $g^{ij}$. In what follows I use the abstract index notations, so square brackets (resp. parentheses) denote anti-symmetrization (resp. symmetrization) over the enclosed indices.
For any metric $(L_{X}g)_{ij} = 2D_{(i}X_{j)}$ and $$\label{e1} (L_{X}^{2}g)_{ij} = X^{p}D_{p}(L_{X}g)_{ij} + (D_{i}X^{p})(L_{X}g)_{pj} + (D_{j}X^{p})(L_{X}g)_{ip}.$$ Trace this to obtain $$\label{e2} g^{ij}(L_{X}^{2}g)_{ij} = 2X^{p}D_{p}D^{q}X_{q} + 4D^{(p}X^{q)}D_{(p}X_{q)}.$$ By assumption this equals $g^{ij}(L_{Y}g)_{ij} = 2D^{p}Y_{p}$. Since by assumption $M$ is compact and without boundary, integration by parts yields \begin{align}\label{e3} 0 = 4\int_{M}D^{(p}X^{q)}D_{(p}X_{q)} - 2\int_{M}(D_{p}X^{p})^{2}. \end{align} In general, this yields no obvious conclusions. Go back to the equation preceeding the integration and commute derivatives to obtain \begin{align}\label{e4} g^{ij}(L_{X}^{2}g)_{ij} = 2X^{p}D^{q}D_{p}X_{q} + 4D^{(p}X^{q)}D_{(p}X_{q)} -2R_{pq}X^{p}X^{q}, \end{align} in which $R_{ij}$ is the Ricci curvature of $g_{ij}$. Now integrating by parts yields \begin{align}\label{e5} \begin{split}0 &= \int_{M}\left(4D^{(p}X^{q)}D_{(p}X_{q)} - 2D^{q}X^{p}D_{p}X_{q} - 2R_{pq}X^{p}X^{q}\right)\\ & = 2\int_{M}\left(D^{(p}X^{q)}D_{(p}X_{q)} + D^{[p}X^{q]}D_{[p}X_{q]} -R_{pq}X^{p}X^{q}\right). \end{split} \end{align} Because $g$ is Riemannian, if the Ricci curvature is non-positive this implies that $D_{(i}X_{j)} = 0$ and $D_{[i}X_{j]} = 0$ so that $D_{i}X_{j} = 0$ and $X$ is parallel. In the original equation this implies $Y$ is Killing. By the original Bochner argument, if the Ricci curvature is non-positive and somewhere negative then there is no non-zero Killing field, so the only possibility for non-trivial solutions is that $g$ be Ricci flat, in which case the Bochner argument forces $Y$ to be parallel.