  Geometry and topology of decision regions are closely related with
classification performance and robustness against adversarial attacks. In this
paper, we use differential geometry to theoretically explore the geometrical
and topological properties of decision regions produced by deep neural networks
(DNNs). The goal is to obtain some geometrical and topological properties of
decision boundaries for given DNN models, and provide some principled guidance
to design and regularization of DNNs. First, we present the curvatures of
decision boundaries in terms of network parameters, and give sufficient
conditions on network parameters for producing flat or developable decision
boundaries. Based on the Gauss-Bonnet-Chern theorem in differential geometry,
we then propose a method to compute the Euler characteristics of compact
decision boundaries, and verify it with experiments.
