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Lyusternik–Fet Theorem
In mathematics, the Lyusternik–Fet theorem states that on every compact Riemannian manifold there exists a closed geodesic. It is named after Lazar Lyusternik and Abram Ilyich Fet Abram Fet (russian: Абрам Ильич Фет) (5 December 1924 — 30 July 2009) was a Russian mathematician, Soviet dissident, philosopher, Samizdat translator and writer. He used various pseudonyms for Samizdat, like N. A. Klenov, A.B. N .... References * https://www.encyclopediaofmath.org/index.php/Closed_geodesic * L.A. Lyusternik, A.I. Fet, "Variational problems on closed manifolds" Dokl. Akad. Nauk. SSSR, 81 (1951) pp. 17–18 (In Russian) Differential geometry Geodesic (mathematics) {{DEFAULTSORT:Lyusternik-Fet theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a metric tensor, Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be Smoothness, smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz continuity, Lipschitz Riemannian metrics or Measurable function, measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Geodesic
In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic, geodesic flow on the tangent space of the manifold. Definition In a Riemannian manifold (''M'',''g''), a closed geodesic is a curve \gamma:\mathbb R\rightarrow M that is a geodesic for the metric ''g'' and is periodic. Closed geodesics can be characterized by means of a variational principle. Denoting by \Lambda M the space of smooth 1-periodic curves on ''M'', closed geodesics of period 1 are precisely the critical point (mathematics), critical points of the energy function E:\Lambda M\rightarrow\mathbb R, defined by : E(\gamma)=\int_0^1 g_(\dot\gamma(t),\dot\gamma(t))\,\mathrmt. If \gamma is a closed geodesic of period ''p'', the reparametrized curve t\mapsto\gamma(pt) is a closed geodesic of period 1, and therefore it is a critical poi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazar Lyusternik
Lazar Aronovich Lyusternik (also Lusternik, Lusternick, Ljusternik; ; 31 December 1899, in Zduńska Wola, Congress Poland, Russian Empire – 23 July 1981, in Moscow, Soviet Union) was a Soviet mathematician. He is famous for his work in topology and differential geometry, to which he applied the variational principle. Using the theory he introduced, together with Lev Schnirelmann, he proved the theorem of the three geodesics, a conjecture by Henri Poincaré that every convex body in 3-dimensions has at least three simple closed geodesics. The ellipsoid with distinct but nearly equal axis is the critical case with exactly three closed geodesics. The ''Lusternik–Schnirelmann theory'', as it is called now, is based on the previous work by Poincaré, David Birkhoff, and Marston Morse. It has led to numerous advances in differential geometry and topology. For this work Lyusternik received the Stalin Prize in 1946. In addition to serving as a professor of mathematics at Moscow St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abram Ilyich Fet
Abram Fet (russian: Абрам Ильич Фет) (5 December 1924 — 30 July 2009) was a Russian mathematician, Soviet dissident, philosopher, Samizdat translator and writer. He used various pseudonyms for Samizdat, like N. A. Klenov, A.B. Nazyvayev, D.A. Rassudin, S.T. Karneyev, etc. If published, his translations were usually issued under the name of A.I. Fedorov, which reproduced Fet's own initials and sometimes under the names of real people who agreed to publish Fet's translations under their names. Biography Abram Fet was born on 5 December 1924 in Odessa into a family of Ilya Fet and Revekka Nikolayevskaya. Ilya Fet was a medical doctor; he was born and grew in Rovno and studied medicine in Paris. Revekka was a housewife; she grew in Odessa. Fet's father often changed jobs, moving with his family over Ukraine looking for places where to escape starvation, and the children had to change schools. In 1936, the family settled in Odessa. There Abram Fet finished high scho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
