In this paper we show how
nonassociative algebras over the real numbers arise from multiplications on certain
homogeneous spaces; that is, an analytic function μ : M × M → M. Then these
algebras are used to obtain an invariant connection ∇ on the homogeneous space and
we give some applications of nonassociative algebras to these topics. Conversely every
finite dimensional nonassociative algebra over the real numbers arises from an
invariant connection and a local multiplication on a homogeneous space. Thus,
analogous to the theory of Lie groups and Lie algebras, much of the basic theory of
nonassociative algebras can be formulated in terms of multiplications and connections
and conversely.
