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Geodesic Manifold
In mathematics, a complete manifold (or geodesically complete manifold) is a ( pseudo-) Riemannian manifold for which, starting at any point , there are straight paths extending infinitely in all directions. Formally, a manifold M is (geodesically) complete if for any maximal geodesic \ell : I \to M, it holds that I=(-\infty,\infty). A geodesic is maximal if its domain cannot be extended. Equivalently, M is (geodesically) complete if for all points p \in M, the exponential map at p is defined on T_pM, the entire tangent space at p. Hopf–Rinow theorem The Hopf–Rinow theorem gives alternative characterizations of completeness. Let (M,g) be a ''connected'' Riemannian manifold and let d_g : M \times M \to Complete metric space">complete (every d_g-Cauchy sequence converges), * All closed and bounded subsets of M are compact. Examples and non-examples Euclidean space \mathbb^n, the n-sphere, sphere \mathbb^n, and the torus, tori \mathbb^n (with their natural Riemannian m ... [...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] |
Homogeneous Space
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ''G'' are called the symmetries of ''X''. A special case of this is when the group ''G'' in question is the automorphism group of the space ''X'' – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, ''X'' is homogeneous if intuitively ''X'' looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of ''G'' be faithful (non-identity elements act non-trivially), althou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic (mathematics)
In geometry, a geodesic () is a curve representing in some sense the locally shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection (mathematics), connection. It is a generalization of the notion of a "Line (geometry), straight line". The noun ''wikt:geodesic, geodesic'' and the adjective ''wikt:geodetic, 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 geodesic, ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's Planetary surface, surface. For a spherical Earth, it is a line segment, 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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal completed the next year, after a regulatory review. Thus, Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Psychology is the scientific study of mind and behavior. Its subject matter includes the behavior of humans and nonhumans, both consciousness, conscious and Unconscious mind, unconscious phenomena, and mental processes such as thoughts, feel ... Well-known products include the '' Methods in Enzymology'' series and encyclopedias such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Penrose–Hawking Singularity Theorems
The Penrose–Hawking singularity theorems (after Roger Penrose and Stephen Hawking) are a set of results in general relativity that attempt to answer the question of when gravitation produces singularities. The Penrose singularity theorem is a theorem in semi-Riemannian geometry and its general relativistic interpretation predicts a gravitational singularity in black hole formation. The Hawking singularity theorem is based on the Penrose theorem and it is interpreted as a gravitational singularity in the Big Bang situation. Penrose shared half of the Nobel Prize in Physics in 2020 "for the discovery that black hole formation is a robust prediction of the general theory of relativity". Singularity A singularity in solutions of the Einstein field equations is one of three things: * Spacelike singularities: The singularity lies in the future or past of all events within a certain region. The Big Bang singularity and the typical singularity inside a non-rotating, uncharge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big Bang
The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including the abundance of light elements, the cosmic microwave background (CMB) radiation, and large-scale structure. The uniformity of the universe, known as the horizon and flatness problems, is explained through cosmic inflation: a phase of accelerated expansion during the earliest stages. A wide range of empirical evidence strongly favors the Big Bang event, which is now essentially universally accepted.: "At the same time that observations tipped the balance definitely in favor of the relativistic big-bang theory, ..." Detailed measurements of the expansion rate of the universe place the Big Bang singularity at an estimated billion years ago, which is considered the age of the universe. Extrapolating this cosmic expansion backward in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarzschild Metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916. According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has neither electric charge nor angular momentum (non-rotating). A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General theory of relativity, relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time in physics, time, or four-dimensional spacetime. In particular, the ''curvature of spacetime'' is directly related to the energy and momentum of whatever is present, including matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations. Newton's law of universal gravitation, which describes gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifton–Pohl Torus
In geometry, the Clifton–Pohl torus is an example of a compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete (by the Hopf–Rinow theorem), this space shows that the same implication does not generalize to pseudo-Riemannian manifolds.. It is named after Yeaton H. Clifton and William F. Pohl, who described it in 1962 but did not publish their result.. Definition Consider the manifold \mathrm = \mathbb^2 \setminus \ with the metric :g= \frac Any homothety is an isometry of M, in particular including the map: :\lambda(x,y)=(2x, 2y) Let \Gamma be the subgroup of the isometry group generated by \lambda. Then \Gamma has a proper, discontinuous action on M. Hence the quotient T = M/\Gamma, which is topologically the torus, is a Lorentz surface that is called the Clifton–Pohl torus. Sometimes, by extension, a surface is called a Clifton–Pohl torus if it is a finite covering of the quotient of M by any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Punctured Plane Is Not Geodesically Complete
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Puncture, punctured or puncturing may refer to: * a flat tyre in British English (US English "flat tire" or just "flat") * a penetrating wound caused by pointy objects as nails or needles * Lumbar puncture, also known as a spinal tap * Puncture (band), an English punk band * ''Puncture'' (film), a 2011 American film starring Chris Evans * Puncture (topology), the removal of a finite set of points from a manifold * Punctured neighbourhood, in general topology * in coding theory, a punctured code, in which some of the bits of the data stream have been removed * Pneumothorax, also known as punctured lung See also * Punctuality * Punctuation Punctuation marks are marks indicating how a piece of writing, written text should be read (silently or aloud) and, consequently, understood. The oldest known examples of punctuation marks were found in the Mesha Stele from the 9th century BC, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (''M'', ''g'') is said to be symmetric if and only if, for each point ''p'' of ''M'', there exists an isometry of ''M'' fixing ''p'' and acting on the tangent space T_pM as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
