In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed
In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed
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 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 categorically.
In geometric topology a basic type are embeddings, of which knot theory is a central example, and generalizations such as immersions, submersions, covering spaces, and ramified covering spaces. Basic results include the Whitney embedding theorem and Whitney immersion theorem.
![Riemann surface for the function f(z) = [?]z, shown as a ramified covering space of the complex plane. Riemann sqrt.svg](http://upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Riemann_sqrt.svg/220px-Riemann_sqrt.svg.png)
In complex geometry, ramified covering spaces are used to model Riemann surfaces, and to analyze maps between surfaces, such as by the Riemann–Hurwitz formula.
In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of isometric embeddings, isometric immersions, and Riemannian submersions; a basic result is the Nash embedding theorem.
A basic example of maps between manifolds are scalar-valued functions on a manifold, or sometimes called regular functions or functionals, by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold.
In geometric topology, most commonly studied are Morse functions, which yield handlebody decompositions, which generalize to Morse–Bott functions and can be used for instance to understand classical groups, such as in Bott periodicity.
In mathematical analysis, one often studies solution to partial differential equations, an important example of which is harmonic analysis, where one studies harmonic functions: the kernel of the Laplace operator. This leads to such functions as the spherical harmonics, and to heat kernel methods of studying manifolds, such as hearing the shape of a drum and some proofs of the Atiyah–Singer index theorem.
The monodromy around a singularity or branch point is an important part of analyzing such functions.
Dual to scalar-valued functions – maps – are maps which correspond to curves or paths in a manifold. One can also define these where the domain is an interval especially the unit interval or where the domain is a circle (equivalently, a periodic path) S1, which yields a loop. These are used to define the fundamental group, chains in homology theory, geodesic curves, and systolic geometry.
Embedded paths and loops lead to knot theory, and related structures such as links, braids, and tangles.
Riemannian manifolds are special cases of metric spaces, and thus one has a notion of Lipschitz continuity, Hölder condition, together with a coarse structure, which leads to notions such as coarse maps and connections with geometric group theory.
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology.
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
In differential geometry, a Riemannian manifold or Riemannian space(M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p.
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.
In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion.
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.
In mathematics, the homotopy principle is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as the immersion problem, isometric immersion problem, fluid dynamics, and other areas.
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
Richard Streit Hamilton is an American mathematician who serves as the Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. Hamilton is best known for foundational contributions to the theory of the Ricci flow and the development of a corresponding program of techniques and ideas for resolving the Poincaré conjecture and geometrization conjecture in the field of geometric topology. Grigori Perelman built upon Hamilton's results to prove the conjectures, and was awarded a Millennium Prize for his work. However, Perelman declined the award, regarding Hamilton's contribution as being equal to his own.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting. Analogously to the finite-dimensional situation, one can define a differentiable Hilbert manifold by considering a maximal atlas in which the transition maps are differentiable.
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.
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory. More recently, it refers largely to the use of nonlinear partial differential equations to study geometric and topological properties of spaces, such as submanifolds of Euclidean space, Riemannian manifolds, and symplectic manifolds. This approach dates back to the work by Tibor Radó and Jesse Douglas on minimal surfaces, John Forbes Nash Jr. on isometric embeddings of Riemannian manifolds into Euclidean space, work by Louis Nirenberg on the Minkowski problem and the Weyl problem, and work by Aleksandr Danilovich Aleksandrov and Aleksei Pogorelov on convex hypersurfaces. In the 1980s fundamental contributions by Karen Uhlenbeck, Clifford Taubes, Shing-Tung Yau, Richard Schoen, and Richard Hamilton launched a particularly exciting and productive era of geometric analysis that continues to this day. A celebrated achievement was the solution to the Poincaré conjecture by Grigori Perelman, completing a program initiated and largely carried out by Richard Hamilton.
In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f : M → N is an immersion if
In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain.
