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Maps Of Manifolds
In mathematics, more specifically in differential geometry and topology, various types of Map (mathematics), maps or Function (mathematics), functions between manifolds are studied, both as objects in their own right and for the light they shed. Types of maps Just as there are various types of manifolds, there are various types of maps of manifolds. In geometric topology, the basic types of maps correspond to various category (mathematics), categories of manifolds: DIFF for smooth functions between differentiable manifolds, PL for piecewise linear functions between piecewise linear manifolds, and TOP for continuous functions between topological manifolds. These are progressively weaker structures, properly connected via PDIFF, the category of piecewise-smooth maps between piecewise-smooth manifolds. In addition to these general categories of maps, there are maps with special properties; these may or may not form categories, and may or may not be generally discussed categorical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometric Immersion
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map f:X\rightarrow Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map f:X\rightarrow Y is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Hurwitz Formula
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves. Statement For a compact, connected, orientable surface S, the Euler characteristic \chi(S) is :\chi(S)=2-2g, where ''g'' is the genus (the ''number of handles''). This follows, as the Betti numbers are 1, 2g, 1, 0, 0, \dots. For the case of an (''unramified'') covering map of surfaces :\pi\colon S' \to S that is surjective and of degree N, we have the formula :\chi(S') = N\cdot\chi(S). That is because each simplex of S should be covered by exactly N in S', at least if we use a fine enough triangulation of S, as we are entitled to do since the Euler ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include Graph of a function, graphs of Multivalued function, multivalued functions such as √''z'' or log(''z''), e.g. the subset of pairs with . Every Riemann surface is a Surface (topology), surface: a two-dimensional real manifold, but it contains more structure (specifically a Complex Manifold, complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and Metrizabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Sqrt
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time. Early years Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Immersion Theorem
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m>1, any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space, and a (not necessarily one-to-one) immersion in (2m-1)-space. Similarly, every smooth m-dimensional manifold can be immersed in the 2m-1-dimensional sphere (this removes the m>1 constraint). The weak version, for 2m+1, is due to transversality ( general position, dimension counting): two ''m''-dimensional manifolds in \mathbf^ intersect generically in a 0-dimensional space. Further results William S. Massey went on to prove that every ''n''-dimensional manifold is cobordant to a manifold that immerses in S^ where a(n) is the number of 1's that appear in the binary expansion of n. In the same paper, Massey proved that for every ''n'' there is manifold (which happens to be a product of real projective spaces) that does not immerse in S^. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Embedding Theorem
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: *The strong Whitney embedding theorem states that any smooth real - dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real -space, if . This is the best linear bound on the smallest-dimensional Euclidean space that all -dimensional manifolds embed in, as the real projective spaces of dimension cannot be embedded into real -space if is a power of two (as can be seen from a characteristic class argument, also due to Whitney). *The weak Whitney embedding theorem states that any continuous function from an -dimensional manifold to an -dimensional manifold may be approximated by a smooth embedding provided . Whitney similarly proved that such a map could be approximated by an immersion provided . This last result is sometimes called the Whitney immersion theorem. About the proof Weak embedding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramified Covering Space
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. In complex analysis In complex analysis, the basic model can be taken as the ''z'' → ''z''''n'' mapping in the complex plane, near ''z'' = 0. This is the standard local picture in Riemann surface theory, of ramification of order ''n''. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus. In algebraic topology In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The ''z'' → ''z''''n'' mapping shows this as a local pattern: if we exclude ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covering Space
In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If p : \tilde X \to X is a covering, (\tilde X, p) is said to be a covering space or cover of X, and X is said to be the base of the covering, or simply the base. By abuse of terminology, \tilde X and p may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étalé space. Covering spaces first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Bernhard Riemann, Riemann as domains on which naturally multivalued function, multivalued complex functions become single-valued. These spaces are now called Riemann surfaces. Covering spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose pushforward (differential), differential is everywhere surjective. It is a basic concept in differential topology, dual to that of an immersion (mathematics), immersion. Definition Let ''M'' and ''N'' be differentiable manifolds, and let f\colon M\to N be a differentiable map between them. The map is a submersion at a point p \in M if its pushforward (differential), differential :Df_p \colon T_p M \to T_N is a surjective linear map. In this case, is called a regular point of the map ; otherwise, is a critical point (mathematics), ''critical point''. A point q \in N is a regular value of if all points in the preimage f^(q) are regular points. A differentiable map that is a submersion at each point p \in M is called a submersion. Equivalently, is a submersion if its differential Df_p has rank (differential topology), constant rank equal to the dimension of . Some authors use the ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
