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Toponogov Theorem
In the mathematical field of Riemannian geometry, Toponogov's theorem (named after Victor Andreevich Toponogov) is a triangle comparison theorem. It is one of a family of comparison theorems that quantify the assertion that a pair of geodesics emanating from a point ''p'' spread apart more slowly in a region of high curvature than they would in a region of low curvature. Let ''M'' be an ''m''-dimensional Riemannian manifold with sectional curvature ''K'' satisfying K\ge \delta\,. Let ''pqr'' be a geodesic triangle, i.e. a triangle whose sides are geodesics, in ''M'', such that the geodesic ''pq'' is minimal and if δ > ''0'', the length of the side ''pr'' is less than \pi / \sqrt \delta. Let ''p''′''q''′''r''′ be a geodesic triangle in the model space ''M''δ, i.e. the simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smoothly from point to point). This gives, in particular, local notions of angle, arc length, length of curves, surface area and volume. From those, some other global quantities can be derived by integral, integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in Three-dimensional space, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Andreevich Toponogov
Victor Andreevich Toponogov (; March 6, 1930 – November 21, 2004) was an outstanding Russian mathematician, noted for his contributions to differential geometry and so-called Riemannian geometry "in the large". Biography After finishing secondary school in 1948, Toponogov entered the department of Mechanics and Mathematics at Tomsk State University, graduated with honours in 1953, and continued as a graduate student there until 1956. He moved to an institution in Novosibirsk in 1956 and lived in that city for the rest of his career. Since the institution at Novosibirsk had not yet been fully credentialed, he had defended his Ph.D. thesis at Moscow State University in 1958, on a subject in Riemann spaces. Novosibirsk State University was established in 1959. In 1961, Toponogov became a professor at a newly created Institute of Mathematics and Computing in Novosibirsk affiliated with the state university. Toponogov's scientific interests were influenced by his advisor A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Theorem
In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry. Differential equations In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof), provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property. Differential (or integral) inequalities, derived from differential (respectively, integral) equations by replacing the equality sign with an inequality sign, form a broad class of such auxiliary relations. One instance of such theorem was used by Aronson and Weinberger to characterize solutions of Fisher's equation, a reaction-diffusion equation. Other examples of comparison theorems include: * Chaplygin's theorem * Grönwall's inequality, and its various generalizations, prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifold, manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sectional Curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a point ''p'' of the manifold. It can be defined geometrically as the Gaussian curvature of the surface (topology), surface which has the plane σ''p'' as a tangent plane at ''p'', obtained from geodesics which start at ''p'' in the directions of σ''p'' (in other words, the image of σ''p'' under the exponential map (Riemannian geometry), exponential map at ''p''). The sectional curvature is a real-valued function on the 2-Grassmannian fiber bundle, bundle over the manifold. The sectional curvature determines the Riemann curvature tensor, Riemann curvature tensor completely. Definition Given a Riemannian manifold and two linearly independent tangent vectors at the same point, ''u'' and ''v'', we can define :K(u,v)= Here ''R' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic Triangle
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a " straight line". The noun ''geodesic'' and the adjective '' geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any Loop (topology), loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Curvature
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point. For example, a sphere is a surface of constant positive curvature. Classification The Riemannian manifolds of constant curvature can be classified into the following three cases: * Elliptic geometry – constant positive sectional curvature * Euclidean geometry – constant vanishing sectional curvature * Hyperbolic geometry – constant negative sectional curvature. Properties * Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel \nabla \mathrm=0. * Every space of constant curvature is locally maximally symmetric, i.e. it has \frac n (n+1) number of local isometries, where ... [...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 \wideti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Riemannian Geometry
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
