A geometric point of view on biderivations

In this paper, we extend the notion of tangent vectors at a point of a smooth manifold (also known as derivations) by introducing biderivations on manifolds. This is a geometric counterpart of the algebraic notion of biderivation, which has been extensively studied in the beginning in rings, and subsequently in the context of non-associative algebra. Our construction leads to the definition of a bitangent space at a pair of points. When the construction is carried out on a single manifold and the two points coincide, this object reduces to a contravariant 2-tensor. We then introduce the bidifferential of a pair of smooth maps as a linear map between bitangent spaces, and we describe its associated matrix explicitly. Finally, we illustrate the theory through applications to some partial differential equations. Indeed, we discuss connections with the eikonal equation, a classical problem arising in the study of wave propagation, and elliptic operators.