Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark Triangle and Beyond Xingwang Xu Nanjing University October 10, 2018 Xingwang Xu Geometry, Analysis and Topology 2 Chern-Gaussian-Bonnet formula for Riemannian manifolds 3 Generalization to complete Riemann surfaces 4 Many attempts for higher dimensional complete manifolds 5 My observation 6 Final Remarks Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark 1 A formula for compact surfaces Xingwang Xu Geometry, Analysis and Topology 3 Generalization to complete Riemann surfaces 4 Many attempts for higher dimensional complete manifolds 5 My observation 6 Final Remarks Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark 1 A formula for compact surfaces 2 Chern-Gaussian-Bonnet formula for Riemannian manifolds Xingwang Xu Geometry, Analysis and Topology 4 Many attempts for higher dimensional complete manifolds 5 My observation 6 Final Remarks Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark 1 A formula for compact surfaces 2 Chern-Gaussian-Bonnet formula for Riemannian manifolds 3 Generalization to complete Riemann surfaces Xingwang Xu Geometry, Analysis and Topology 5 My observation 6 Final Remarks Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark 1 A formula for compact surfaces 2 Chern-Gaussian-Bonnet formula for Riemannian manifolds 3 Generalization to complete Riemann surfaces 4 Many attempts for higher dimensional complete manifolds Xingwang Xu Geometry, Analysis and Topology 6 Final Remarks Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark 1 A formula for compact surfaces 2 Chern-Gaussian-Bonnet formula for Riemannian manifolds 3 Generalization to complete Riemann surfaces 4 Many attempts for higher dimensional complete manifolds 5 My observation Xingwang Xu Geometry, Analysis and Topology Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark 1 A formula for compact surfaces 2 Chern-Gaussian-Bonnet formula for Riemannian manifolds 3 Generalization to complete Riemann surfaces 4 Many attempts for higher dimensional complete manifolds 5 My observation 6 Final Remarks Xingwang Xu Geometry, Analysis and Topology The sum of the exterior angles of a triangle equals 2π The sum of the interior angles of a triangle equals π χ(∆) := The number of surfaces - the number of edges + the number of vertexes equal 1 2πχ(∆) = the sum of the exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces triangle C For a triangle: AB Xingwang Xu Geometry, Analysis and Topology The sum of the interior angles of a triangle equals π χ(∆) := The number of surfaces - the number of edges + the number of vertexes equal 1 2πχ(∆) = the sum of the exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces triangle C For a triangle: AB The sum of the exterior angles of a triangle equals 2π Xingwang Xu Geometry, Analysis and Topology χ(∆) := The number of surfaces - the number of edges + the number of vertexes equal 1 2πχ(∆) = the sum of the exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces triangle C For a triangle: AB The sum of the exterior angles of a triangle equals 2π The sum of the interior angles of a triangle equals π Xingwang Xu Geometry, Analysis and Topology 2πχ(∆) = the sum of the exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces triangle C For a triangle: AB The sum of the exterior angles of a triangle equals 2π The sum of the interior angles of a triangle equals π χ(∆) := The number of surfaces - the number of edges + the number of vertexes equal 1 Xingwang Xu Geometry, Analysis and Topology Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces triangle C For a triangle: AB The sum of the exterior angles of a triangle equals 2π The sum of the interior angles of a triangle equals π χ(∆) := The number of surfaces - the number of edges + the number of vertexes equal 1 2πχ(∆) = the sum of the exterior angles Xingwang Xu Geometry, Analysis and Topology The sum of the exterior angles of a polygon equals 2π The sum of the interior angles of a polygon equals (n − 2)π where n is the number of the edges χ(polygon) = the number of vertexes - the number of edges + the number of faces = 1 Add more edges, i.e., to do triangulation (easy to see: χ does not change.) 2πχ(polygon) = the sum of the exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces polygon C D For a polygon: AB Xingwang Xu Geometry, Analysis and Topology The sum of the interior angles of a polygon equals (n − 2)π where n is the number of the edges χ(polygon) = the number of vertexes - the number of edges + the number of faces = 1 Add more edges, i.e., to do triangulation (easy to see: χ does not change.) 2πχ(polygon) = the sum of the exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces polygon C D For a polygon: AB The sum of the exterior angles of a polygon equals 2π Xingwang Xu Geometry, Analysis and Topology χ(polygon) = the number of vertexes - the number of edges + the number of faces = 1 Add more edges, i.e., to do triangulation (easy to see: χ does not change.) 2πχ(polygon) = the sum of the exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces polygon C D For a polygon: AB The sum of the exterior angles of a polygon equals 2π The sum of the interior angles of a polygon equals (n − 2)π where n is the number of the edges Xingwang Xu Geometry, Analysis and Topology Add more edges, i.e., to do triangulation (easy to see: χ does not change.) 2πχ(polygon) = the sum of the exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces polygon C D For a polygon: AB The sum of the exterior angles of a polygon equals 2π The sum of the interior angles of a polygon equals (n − 2)π where n is the number of the edges χ(polygon) = the number of vertexes - the number of edges + the number of faces = 1 Xingwang Xu Geometry, Analysis and Topology 2πχ(polygon) = the sum of the exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces polygon C D For a polygon: AB The sum of the exterior angles of a polygon equals 2π The sum of the interior angles of a polygon equals (n − 2)π where n is the number of the edges χ(polygon) = the number of vertexes - the number of edges + the number of faces = 1 Add more edges, i.e., to do triangulation (easy to see: χ does not change.) Xingwang Xu Geometry, Analysis and Topology Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces polygon C D For a polygon: AB The sum of the exterior angles of a polygon equals 2π The sum of the interior angles of a polygon equals (n − 2)π where n is the number of the edges χ(polygon) = the number of vertexes - the number of edges + the number of faces = 1 Add more edges, i.e., to do triangulation (easy to see: χ does not change.) 2πχ(polygon) = the sum of the exterior angles Xingwang Xu Geometry, Analysis and Topology The total sum of the infinitesimal exterior angles equals 2π No definition of the interior angles χ(disc) = the number of vertexes - the number of edges + the number of faces = 1 (by triangulation) 2πχ(Disc) = the sum of the infinitesimal exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces circle For a disc: '$- p &% Xingwang Xu Geometry, Analysis and Topology No definition of the interior angles χ(disc) = the number of vertexes - the number of edges + the number of faces = 1 (by triangulation) 2πχ(Disc) = the sum of the infinitesimal exterior angles Outline Triangle Surfaces Regular Polygon Riemmanian Manifolds Circle Complete manifolds Annulus Domain New Development General domain Final Remark Compact Riemann Surfaces circle For a disc: '$- p &% The total sum of the infinitesimal exterior angles equals 2π Xingwang Xu Geometry, Analysis and Topology χ(disc) = the number of vertexes - the number of edges + the number of faces = 1 (by triangulation) 2πχ(Disc) = the sum of the infinitesimal exterior angles Outline Triangle Surfaces Regular