  Convolutions are fundamental elements in deep learning architectures. Here,
we present a theoretical framework for combining extrinsic and intrinsic
approaches to manifold convolution through isometric embeddings into tori. In
this way, we define a convolution operator for a manifold of arbitrary topology
and dimension. We also explain geometric and topological conditions that make
some local definitions of convolutions which rely on translating filters along
geodesic paths on a manifold, computationally intractable. A result of Alan
Turing from 1938 underscores the need for such a toric isometric embedding
approach to achieve a global definition of convolution on computable, finite
metric space approximations to a smooth manifold.
