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Splitting Theorem
In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of Lorentzian manifolds. Cheeger and Gromoll's Riemannian splitting theorem Any connected Riemannian manifold has an underlying metric space structure, and this allows the definition of a ''geodesic line'' as a map such that the distance from to equals for arbitrary and . This is to say that the restriction of to any bounded interval is a curve of minimal length which connects its endpoints. In 1971, Jeff Cheeger and Detlef Gromoll proved that, if a geodesically complete and connected Riemannian manifold of nonnegative Ricci curvature contains any geodesic line, then it must split isometrically as the product of a complete Riemannian manifold with . The proof was later simplifi ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Maximum Principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. In the simplest case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unless is a constant function, the maximum cannot also be achieved anywhere on itself. Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the size ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book form called ''Surveys in Differential Geometry''. It covers differential geometry and related subjects such as differential equations, mathematical physics, algebraic geometry, and geometric topology. The editor-in-chief is Shing-Tung Yau of Harvard University. History The journal was established in 1967 by Chuan-Chih Hsiung, who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journal's editor-in-chief, and later co-editor-in-chief, until his death in 2009. In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Globally Hyperbolic Manifold
In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is * t^2 - r^2 = T^2 (t and r being the usual variables of time and radius) which is one of the usual equations representing an hyperbola. But this expression is only true relative to the ordinary origin; this article then outline bases for generalizing the concept to any pair of points in spacetime. This is relevant to Albert Einstein's theory of general relativity, and potentially to other metric gravitational theories. Definitions There are several equivalent definitions of global hyperbolicity. Let ''M'' be a smooth connected Lorentzian manifold without boundary. We make the following preliminary definitions: * ''M'' is ''non-totally vicious'' if there is at least one point such that no closed timelike curve passes through it. * ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathematics at Tsinghua University. Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Manifold
In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°. The universal cover of a complete flat manifold is Euclidean space. This can be used to prove the theorem of that all compact flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by . Examples The following manifolds can be endowed with a flat metric. Note that this may not be their 'standard' metric (for example, the flat metric on the 2-dimensional torus is not the metric induced by its usual embedding into \mathbb^3). Dimension 1 Every one-dimensional Riemannian manifold is flat. Conversely, given that every connected one-dimensional smooth manifold is diffeomorphic to either \mathbb or S^1, it is straightforward to see that every connected one-dimensional Riemannian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space. Properties A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a topological space ''X'' the following are all equivalent: *''X'' is contractible (i.e. the identity map is null-homotopic). *''X'' is homotopy equivalent to a one-point space. *''X'' deformation retracts onto a point. (However, there exist contractible spaces which do not ''strongly'' deformation retract to a point.) *For any space ''Y'', any two maps ''f'','' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the '' Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Manifold
In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. A line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Open manifolds For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact. Abuse of language Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Cover
A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete space D and for every x \in X an open neighborhood U \subset X, such that \pi^(U)= \displaystyle \bigsqcup_ V_d and \pi, _:V_d \rightarrow U is a homeomorphism for every d \in D . Often, the notion of a covering is used for the covering space E as well as for the map \pi : E \rightarrow X. The open sets V_ are called sheets, which are uniquely determined up to a homeomorphism if U is connected. For each x \in X the discrete subset \pi^(x) is called the fiber of x. The degree of a covering is the cardinality of the space D. If E is path-connected, then the covering \pi : E \rightarrow X is denoted as a path-connected covering. Examples * For every topological space X there exists the covering \pi:X \rightarrow X with \pi(x)=x, which is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Submersion
In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Formal definition Let (''M'', ''g'') and (''N'', ''h'') be two Riemannian manifolds and f:M\to N a (surjective) submersion, i.e., a fibered manifold. The horizontal distribution \mathrm(df)^ is a sub-bundle of the tangent bundle of TM which depends both on the projection f and on the metric g. Then, ''f'' is called a Riemannian submersion if and only if the isomorphism df : \mathrm(df)^ \rightarrow TN is an isometry. Examples An example of a Riemannian submersion arises when a Lie group G acts isometrically, freely and properly on a Riemannian manifold (M,g). The projection \pi: M \rightarrow N to the quotient space N = M /G equipped with the quotient metric is a Riemannian submersion. For example, component-wise multiplication on S^3 \subset \mathbb^2 by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holonomy Group
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features. Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy. The most common forms of holonomy are for connections possessing some kind of symmetry. Important examples include: holonomy of the Levi-Civita connection in Riemannian geometry (called Riemannian holonomy), holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of these cases, the holonomy of the connection can be identified with a Lie g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
