# A mathematical background question

$$
\def\hm{{\mathbb{H}M}}
\def\R{{\mathbb{R}}}
$$

## Construction

Consider a Riemannian manifold $M$. Let $F(M)$ denote $L^2$ functions on $M$, and $F(TM)$ denote $L^2$ functions on the tangent bundle.

***Observation 1:*** Using the canonical Riemannian measures on tangent space, $F(TM)$ has a fibrewise $L^2$ inner product with values in $F(M)$, i.e. a map $F(TM) \times F(TM) \to F(M)$.

***Observation 2:*** We can use the exponential map $e : TM \to M$ to pull back functions, $e^\ast : F(M) \to F(TM)$.

Combining these structures, we obtain a multiplication $$ \Delta: F(TM) \times F(M) \to F(M) $$

## Questions

$\Delta$ is a pretty natural operation. It only requires a Riemannian metric and is a kind of generalized, local convolution. Does it have a standard name, or show up anywhere? Maybe something obvious that I'm not thinking of?

In particular, I think it's a generalization of the construction we're playing with. In our case, we'll also have a connection $A$, and a corresonding subspace of $C_A \subset F(TM)$ of functions which are flat (i.e., constant in the horizontal direction) with respect to $A$. Then I think our generalized convolution is exactly the operation

$$ \Delta: C_A \times F(M) \to F(M) $$

right? Does this show up in any pure math context, as such?