Algebraic structures on parallelizable manifolds

Abstract

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map $\rho_p:\mathfrak{l} \longrightarrow T_p \mathbb{L}$ for each point $p \in \mathbb{L}$, where $\mathfrak{l}$ 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 $\mathfrak{l}$. Furthermore, flows of these vector fields give rise to a product between elements of $% \mathfrak{l}$ and $\mathbb{L}$, 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 $\mathfrak{l}$. We will describe the properties and examples of these constructions.