Differential geometry of surfaces

About: Differential geometry of surfaces is a research topic. Over the lifetime, 168 publications have been published within this topic receiving 8308 citations.

Lectures on Differential Geometry

Abstract: Algebraic Preliminaries: 1. Tensor products of vector spaces 2. The tensor algebra of a vector space 3. The contravariant and symmetric algebras 4. Exterior algebra 5. Exterior equations Differentiable Manifolds: 1. Definitions 2. Differential maps 3. Sard's theorem 4. Partitions of unity, approximation theorems 5. The tangent space 6. The principal bundle 7. The tensor bundles 8. Vector fields and Lie derivatives Integral Calculus on Manifolds: 1. The operator $d$ 2. Chains and integration 3. Integration of densities 4. $0$ and $n$-dimensional cohomology, degree 5. Frobenius' theorem 6. Darboux's theorem 7. Hamiltonian structures The Calculus of Variations: 1. Legendre transformations 2. Necessary conditions 3. Conservation laws 4. Sufficient conditions 5. Conjugate and focal points, Jacobi's condition 6. The Riemannian case 7. Completeness 8. Isometries Lie Groups: 1. Definitions 2. The invariant forms and the Lie algebra 3. Normal coordinates, exponential map 4. Closed subgroups 5. Invariant metrics 6. Forms with values in a vector space Differential Geometry of Euclidean Space: 1. The equations of structure of Euclidean space 2. The equations of structure of a submanifold 3. The equations of structure of a Riemann manifold 4. Curves in Euclidean space 5. The second fundamental form 6. Surfaces The Geometry of $G$-Structures: 1. Principal and associated bundles, connections 2. $G$-structures 3. Prolongations 4. Structures of finite type 5. Connections on $G$-structures 6. The spray of a linear connection Appendix I: Two existence theorems Appendix II: Outline of theory of integration on $E^n$ Appendix III: An algebraic model of transitive differential geometry Appendix IV: The integrability problem for geometrical structures References Index.

Differential Geometry of Curves and Surfaces

Abstract: Plane Curves: Local Properties Parametrizations Position, Velocity, and Acceleration Curvature Osculating Circles, Evolutes, and Involutes Natural Equations Plane Curves: Global Properties Basic Properties Rotation Index Isoperimetric Inequality Curvature, Convexity, and the Four-Vertex Theorem Curves in Space: Local Properties Definitions, Examples, and Differentiation Curvature, Torsion, and the Frenet Frame Osculating Plane and Osculating Sphere Natural Equations Curves in Space: Global Properties Basic Properties Indicatrices and Total Curvature Knots and Links Regular Surfaces Parametrized Surfaces Tangent Planes and Regular Surfaces Change of Coordinates The Tangent Space and the Normal Vector Orientable Surfaces The First and Second Fundamental Forms The First Fundamental Form Map Projections (Optional) The Gauss Map The Second Fundamental Form Normal and Principal Curvatures Gaussian and Mean Curvature Developable Surfaces and Minimal Surfaces The Fundamental Equations of Surfaces Gauss's Equations and the Christoffel Symbols Codazzi Equations and the Theorema Egregium The Fundamental Theorem of Surface Theory The Gauss-Bonnet Theorem and Geometry of Geodesics Curvatures and Torsion Gauss-Bonnet Theorem, Local Form Gauss-Bonnet Theorem, Global Form Geodesics Geodesic Coordinates Applications to Plane, Spherical and Elliptic Geometry Hyperbolic Geometry Curves and Surfaces in n-Dimensional Euclidean Space Curves in n-Dimensional Euclidean Space Surfaces in Rn Appendix: Tensor Notation

Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

Abstract: This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

Elementary Differential Geometry

Abstract: Curves in the plane and in space.- How much does a curve curve?.- Global properties of curves.- Surfaces in three dimensions.- Examples of surfaces.- The first fundamental form.- Curvature of surfaces.- Gaussian, mean and principal curvatures.- Geodesics.- Gauss' Theorema Egregium.- Hyperbolic geometry.- Minimal surfaces.- The Gauss-Bonnet theorem.