Linear representations of manifolds

Abstract

A finite-dimensional linear representation of a group or an algebra may be regarded as a map into a space of matrices, endowing abstract elements with coordinates, and encoding algebraic operations as matrix products. With this in mind, we define a linear representation of a $\mathsf{G}$-manifold $\mathcal{M} $ as a map into a space of matrices, representing points as matrices and the $\mathsf{G}$-action as matrix products. We show that this generalizes group representations to any $\mathsf{G}$-manifold that may not have a group structure, with homogeneous spaces $\mathsf{G}/\mathsf{H}$ an important special case; and in this case it also generalizes Cartan embeddings of symmetric spaces to more general $\mathsf{G}/\mathsf{H}$. To demonstrate the utility of such manifold representations, we use them to provide effective bounds for Mostow-Palais $\mathsf{G}$-equivariant embeddings of $\mathsf{G}$-manifolds into $\mathsf{G}$-modules $\mathbb{V}$. Unlike Whitney and Nash embeddings, Mostow-Palais embeddings have no known effective bounds; before our work, it was only known that $\dim \mathbb{V} < \infty$ if $\mathsf{G}$ is compact. We will give explicit values for $\dim \mathbb{V}$ and show that our bounds are sharp. Furthermore, our method is constructive, giving explicit expressions for these minimal-dimensional Mostow-Palais embeddings.