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In differential geometrythe Gauss map named after Carl F. Gauss maps a surface in Euclidean space R 3 to the unit sphere S 2. The Gauss map can be defined globally if and only if the surface is orientablein which case its degree is half the Euler characteristic. The Gauss map can always be defined locally i. The Jacobian determinant of the Gauss map is equal to Gaussian curvatureand the differential of the Gauss map is called the shape operator. There is also a Gauss map for a linkwhich computes linking number.
In this case a point on the submanifold is mapped to its oriented tangent subspace. Finally, the notion of Gauss map can be generalized to an oriented submanifold X of dimension k in an oriented ambient Riemannian manifold M of dimension n. In that case, the Gauss map then goes from X to the set of tangent k -planes in the tangent bundle TM. The area of the image of the Gauss map is called the total curvature and is equivalent to the surface integral of the Gaussian curvature.
This is the original interpretation given by Gauss. The Gauss—Bonnet theorem links total curvature of a surface to its topological properties. The Gauss map reflects many properties of the surface: when the surface has zero Gaussian curvature, that is along a parabolic line the Gauss map will have a fold catastrophe. Both parabolic lines and cusp are stable phenomena and will remain under slight deformations of the surface.
Cusps occur when:. Create your Account. Are you sure you want to cancel your membership with us? Gauss map. This article is about differential geometry. For other uses, see Gauss map disambiguation. This article includes a list of referencesrelated reading or external linksbut its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations.
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Write a review Rate this item: 1 2 3 4 5. Preview this item Preview this item. Series: Research notes in mathematics Subjects Curves on surfaces. Gauss maps. Find a copy online Links to this item Table of contents Table of contents Table of contents.In differential geometrythe Gauss map named after Carl F. Gauss maps a surface in Euclidean space R 3 to the unit sphere S 2. Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Sometimes referred to as the Princeps mathematicorum and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history's most influential mathematicians.
In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space.
For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. The Gauss map can be defined globally if and only if the surface is orientablein which case its degree is half the Euler characteristic. The Gauss map can always be defined locally i.
Singularities of Anti de Sitter torus Gauss maps
The Jacobian determinant of the Gauss map is equal to Gaussian curvatureand the differential of the Gauss map is called the shape operator. In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping.
The degree is always an integer, but may be positive or negative depending on the orientations. In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
It is commonly denoted by. In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature.
There is also a Gauss map for a linkwhich computes linking number. In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component.
Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link. In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space.
Intuitively, the linking number represents the number of times that each curve winds around the other. The linking number is always an integer, but may be positive or negative depending on the orientation of the two curves. In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.
Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally, and sometimes globally. In this case a point on the submanifold is mapped to its oriented tangent subspace. There are different types of submanifolds depending on exactly which properties are required.
Different authors often have different definitions. In mathematics, the Grassmannian Gr kV is a space which parametrizes all k -dimensional linear subspaces of the n-dimensional vector space V.
For example, the Grassmannian Gr 1, V is the space of lines through the origin in Vso it is the same as the projective space of one dimension lower than V. In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid of Alexandria.We study timelike surfaces in Anti de Sitter 3-space as an application of singularity theory.
We define two mappings associated to a timelike surface which are called Anti de Sitter nullcone Gauss image and Anti de Sitter torus Gauss map. We also define a family of functions named Anti de Sitter null height function on the timelike surface. We use this family of functions as a basic tool to investigate the geometric meanings of singularities of the Anti de Sitter nullcone Gauss image and the Anti de Sitter torus Gauss map.
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Rent this article via DeepDyve. Gusein-Zade and A. Singularities of Differentiable Maps vol. Banchoff, T. Gaffney and C. Cusps of Gauss mappings. Research Notes in Mathematics, Google Scholar. Bruce and P. Curvesandsingularities second edition. Cambridge Univ. Press The duals of generic hypersurfaces.
Wavefronts and parallels in Euclidean space. Cambridge Philos. Generic geometry and duality. Singularities Lille,29—59, London Math.
Lecture Note Ser.
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Press, Cambridge Generic geometry, transversality and projections. London Math. On spacelike surfaces in Anti de Sitter 3 -space from the contact view-point. Hokkaido Math. Fusho and S.Publisher : Pitman Advanced Pub. Description : From the table of contents: Gauss mappings of plane curves, Gauss mappings of surfaces, characterizations of Gaussian cusps, singularities of families of mappings, projections to lines, focal and parallel surfaces, projections to planes, singularities and extrinsic geometry.
Download or read it online for free here: Read online online html. Combinatorial Geometry with Application to Field Theory by Linfan Mao - InfoQuest Topics covered in this book include fundamental of mathematical combinatorics, differential Smarandache n-manifolds, combinatorial or differentiable manifolds and submanifolds, Lie multi-groups, combinatorial principal fiber bundles, etc.
Lectures on Exterior Differential Systems by M.
Kuranishi - Tata Institute of Fundamental Research Contents: Parametrization of sets of integral submanifolds Regular linear maps, Germs of submanifolds of a manifold ; Exterior differential systems Differential systems with independent variables ; Prolongation of Exterior Differential Systems.
Lectures on Calabi-Yau and Special Lagrangian Geometry by Dominic Joyce - arXiv An introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture.
Michor - American Mathematical Society This book lays the foundations of differential calculus in infinite dimensions and discusses those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory.We associate weighted graphs to stable Gauss maps on orientable closed surfaces immersed in 3-space and prove that any bipartite weighted graph can be associated to some stable Gauss map.
The Gauss Map, A Dynamical Approach
This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Existence theorems of fold maps.
Japan J. Arnold, S. Gusein-Zade and A. Singularities of differentiable maps. The classification of critical points, caustics and wave fronts. Monographs in Mathematics, Banchoff, T. Gaffney and C. Web version with D. Google Scholar. Bleecker and L. Stability of Gauss maps.
Illinois J. Bruce, P. Giblin and F. Familiesofsurfaces: heightfunctions, Gauss maps and duals. Real and Complex Singularities, ed. Marar, Pitman Research Notes in Mathematics,— International Journal of Computer Vision, 17 3— On singularities of folding type. Golubitsky and V. Stable Mappings and Their Singularities. Springer Verlag, Berlin Guillemin and A. Differential Topology. Prentice-Hall, New Jersey Hacon, C.