Algebraic structures on parallelizable manifolds

Abstract

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold L, there exists a global trivialization of the tangent bundle, which defines a map ρp:l⟶TpL for each point p∈L, where l is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of l. Furthermore, flows of these vector fields give rise to a product between elements of l and L, which in turn induces a local loop structure (i.e. a non-associative analog of a group). Furthermore, we also define a generalization of a Lie algebra structure on l. We will describe the properties and examples of these constructions.