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.　(C)　2021　Elsevier　B.V.　All　rights　reserved.