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Jacobi Field
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi. Definitions and properties Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics \gamma_\tau with \gamma_0=\gamma, then :J(t)=\left.\frac\_ is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic \gamma. A vector field ''J'' along a geodesic \gamma is said to be a Jacobi field if it satisfies the Jacobi equation: :\fracJ(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0, where ''D'' denotes the covariant derivative with respect to the Levi-Civita connection, ''R'' the Riemann curvature tensor, \dot\gamma(t)=d\gamma(t)/dt the tangent vector ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis. Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be ''perpendicular'' if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Katsumi Nomizu
was a Japanese-American mathematician known for his work in differential geometry. Life and career Nomizu was born in Osaka, Japan on the first day of December, 1924. He studied mathematics at Osaka University, graduating in 1947 with a Master of Science then traveled to the United States on a U.S. Army Fulbright Scholarship. He studied first at Columbia University and then at the University of Chicago where in 1953 he became the first student to earn a Ph.D. under the thesis direction of Shiing-Shen Chern. The subject was affine differential geometry, a topic to which he would return much later in his career. He presented his thesis, ''Invariant affine connections on homogeneous spaces'' in 1953. Returning to Japan, he studied at Nagoya University, obtaining a doctor of science in 1955. He published his first volume, ''Lie Groups and Differential Geometry'' dedicated to his wife Kimiko whom he had married that same year. Nomizu taught at Nagoya University until 1958 when ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shoshichi Kobayashi
was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie algebras. Biography Kobayashi graduated from the University of Tokyo in 1953. In 1956, he earned a Ph.D. from the University of Washington under Carl B. Allendoerfer. His dissertation was ''Theory of Connections''. He then spent two years at the Institute for Advanced Study and two years at MIT. He joined the faculty of the University of California, Berkeley in 1962 as an assistant professor, was awarded tenure the following year, and was promoted to full professor in 1966. Kobayashi served as chairman of the Berkeley Mathematics Department for a three-year term from 1978 to 1981 and for the 1992 Fall semester. He chose early retirement under the VERIP plan in 1994. The two-volume book ''Foundations of Differential Geometry'', which he coau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Gregory Ebin
David Gregory Ebin (born 24 October 1942, Los Angeles) is an American mathematician, specializing in differential geometry. Ebin received in 1964 from Harvard University his bachelor's degree and in 1967 his Ph.D. from Massachusetts Institute of Technology under Isadore Singer with thesis ''On the space of Riemannian metrics''. From 1968 to 1969 Ebin was a lecturer at the University of California, Berkeley. He became in 1969 an associate professor and in 1978 a full professor at the Stony Brook University. Ebin in the academic years 1983–1984 and 1991–1992 was a visiting professor at UCLA, in 1971 a docent at the École Polytechnique and the University of Paris VII, and in 1976 a member of the Courant Institute in New York. He was elected a Fellow of the American Mathematical Society in 2012. His research deals with differential geometry, infinite-dimensional manifolds (in hydrodynamics and in his treatment of the space of Riemannian metrics), nonlinear partial differential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeff Cheeger
Jeff Cheeger (born December 1, 1943, Brooklyn, New York City) is a mathematician. Cheeger is professor at the Courant Institute of Mathematical Sciences at New York University in New York City. His main interests are differential geometry and its connections with topology and analysis. Biography Cheeger graduated from Harvard University with a B.A. in 1964. He graduated from Princeton University with an M.S. in 1966 and with a PhD in 1967. He is a Silver Professor at the Courant Institute at New York University where he has worked since 1993. He worked as a teaching assistant and research assistant at Princeton University from 1966–1967, a National Science Foundation postdoctoral fellow and instructor from 1967–1968, an assistant professor from 1968 to 1969 at the University of Michigan, and an associate professor from 1969–1971 at SUNY at Stony Brook. Cheeger was a professor at SUNY, Stony Brook from 1971 to 1985, a leading professor from 1985 to 1990, and a distingu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manfredo Do Carmo
Manfredo Perdigão do Carmo (15 August 1928, Maceió – 30 April 2018, Rio de Janeiro) was a Brazilian mathematician. He spent most of his career at IMPA and is seen as the doyen of differential geometry in Brazil. Education and career Do Carmo studied civil engineering at the University of Recife from 1947 to 1951. After working a few years as engineer, he accepted a teaching position at the newly created Institute of Physics and Mathematics at Recife. On suggestion of Elon Lima, in 1959 he went to Instituto Nacional de Matemática Pura e Aplicada to improve his background and in 1960 he moved to the USA to pursue a Ph.D. in mathematics at the University of California, Berkeley under the supervision of Shiing-Shen Chern. He defended his thesis, entitled "''The Cohomology Ring of Certain Kahlerian Manifolds''", in 1963. After working again at University of Recife and at the University of Brasilia, in 1966 he became professor at Instituto Nacional de Matemática Pura e Aplica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rauch Comparison Theorem
In Riemannian geometry, the Rauch comparison theorem, named after Harry Rauch, who proved it in 1951, is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread. The statement of the theorem involves two Riemannian manifolds, and allows to compare the infinitesimal rate at which geodesics spread apart in the two manifolds, provided that their curvature can be compared. Most of the time, one of the two manifolds is a "comparison model", generally a manifold with constant curvature , and the second one is the manifold under study : a bound (either lower or upper) on its sectional curvature is then needed in order to apply Rauch comparison theorem. Statement Let M, \widetilde be Riemannian manifolds, on which are drawn unit speed geodesic segments \gamma : , T\to M and \widetild ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic Deviation Equation
In general relativity, if two objects are set in motion along two initially parallel trajectories, the presence of a tidal gravitational force will cause the trajectories to bend towards or away from each other, producing a relative acceleration between the objects. Mathematically, the tidal force in general relativity is described by the Riemann curvature tensor, and the trajectory of an object solely under the influence of gravity is called a ''geodesic''. The geodesic deviation equation relates the Riemann curvature tensor to the relative acceleration of two neighboring geodesics. In differential geometry, the geodesic deviation equation is more commonly known as the Jacobi equation. Mathematical definition To quantify geodesic deviation, one begins by setting up a family of closely spaced geodesics indexed by a continuous variable ''s'' and parametrized by an affine parameter τ. That is, for each fixed ''s'', the curve swept out by γ''s''(τ) as τ varies is a geodesic. When ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Points
In differential geometry, conjugate points or focal points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. All geodesics are ''locally'' length-minimizing, but not globally. For example on a sphere, any geodesic passing through the north-pole can be extended to reach the south-pole, and hence any geodesic segment connecting the poles is not (uniquely) ''globally'' length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.Cheeger, Ebin. ''Comparison Theorems in Riemannian Geometry''. North-Holland Publishing Company, 1975, pp. 17-18. Definition Suppose ''p'' and ''q'' are points on a Riemannian manifold, and \gamma is a geodesic that connects ''p'' and ''q''. Then ''p'' and ''q'' are conjugate points along \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Vector Field
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object. Definition Specifically, a vector field ''X'' is a Killing field if the Lie derivative with respect to ''X'' of the metric ''g'' vanishes: :\mathcal_ g = 0 \,. In terms of the Levi-Civita connection, this is :g\left(\nabla_Y X, Z\right) + g\left(Y, \nabla_Z X\right) = 0 \, for all vectors ''Y'' and ''Z''. In local coordinates, this amounts to the Killing equation :\nabla_\mu X_\nu + \nabla_ X_\mu = 0 \,. This condition is expressed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
