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Harmonic Coordinates
In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian metric on the manifold. They are useful in many problems of geometric analysis due to their regularity properties. In two dimensions, certain harmonic coordinates known as isothermal coordinates have been studied since the early 1800s. Harmonic coordinates in higher dimensions were developed initially in the context of Lorentzian geometry and general relativity by Albert Einstein and Cornelius Lanczos (see harmonic coordinate condition). Following the work of Dennis DeTurck and Jerry Kazdan in 1981, they began to play a significant role in the geometric analysis literature, although Idzhad Sabitov and S.Z. Šefel had made the same discovery five years earlier. Definition Let be a Riemannian manifold of dimension . One says that a coordinate chart , defined on an open subset of , is harmonic if each individual coordinate fu ... [...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] |
Harmonic Map
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions. Informally, the Dirichlet energy of a mapping from a Riemannian manifold to a Riemannian manifold can be thought of as the total amount that stretches in allocating each of its elements to a point of . For instance, an unstretched rubber band and a smooth stone can both be naturally viewed as Riemannian manifolds. Any way of stretching the rubber band over the stone can be viewed as a mapping between these manifolds, and the total tension involved is represented by the Dirichlet energy. Harmon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function ''space''. In linear algebra Let be a field and let be any set. The functions → can be given the structure of a vector space over where the operations are defined pointwise, that is, for any , : → , any in , and any in , define \begin (f+g)(x) &= f(x)+g(x) \\ (c\cdot f)(x) &= c\cdot f(x) \end When the domain has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if and also itself are vector spaces over , the se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Manifold
In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic geometry, analytic spaces are a generalization of analytic manifolds such that singularities are permitted. For U \subseteq \R^n, the space of analytic functions, C^(U), consists of infinitely differentiable functions f:U \to \R , such that the Taylor series T_f(\mathbf) = \sum_\frac (\mathbf-\mathbf)^\alpha converges to f(\mathbf) in a neighborhood of \mathbf, for all \mathbf \in U. The requirement that the transition maps be analytic is significantly more restrictive than that they be infinitely differentiable; the analytic manifolds are a proper subset of the smooth, i.e. C^\infty, manifolds. There are many similarities between the theory of analytic and smooth manifolds, but a critical difference is that analytic manifolds do not ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if for every x_0 in its domain, its Taylor series about x_0 converges to the function in some neighborhood of x_0 . This is stronger than merely being infinitely differentiable at x_0 , and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \cdots in which the coefficients a_0, a_1, \dots a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Metric
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to Lorentzian manifolds (including the four-dimensional Lorentzian manifolds usually studied in general relativity). Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons. If M is the underlying n-dimensional manifold, and g is its metric tensor, the Einstein condition means that :\mathrm = kg for some constant k, where \operatorname denotes the Ricci tensor of g. Einstein manifolds with k = 0 are called Ricci-flat manifolds. The Einstein condition and Einstein's equation In local c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schauder Estimates
In mathematics, and more precisely, in Functional analysis and PDEs, the Schauder estimates are a collection of results due to concerning the regularity of solutions to linear, uniformly elliptic partial differential equations. The estimates say that when the equation has appropriately smooth terms and appropriately smooth solutions, then the Hölder norm of the solution can be controlled in terms of the Hölder norms for the coefficient and source terms. Since these estimates assume by hypothesis the existence of a solution, they are called a priori estimates. There is both an ''interior'' result, giving a Hölder condition for the solution in interior domains away from the boundary, and a ''boundary'' result, giving the Hölder condition for the solution in the entire domain. The former bound depends only on the spatial dimension, the equation, and the distance to the boundary; the latter depends on the smoothness of the boundary as well. The Schauder estimates are a necessa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Regularity
In the theory of partial differential equations, a partial differential operator P defined on an open subset :U \subset^n is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty ( smooth), u must also be C^\infty. If this assertion holds with C^\infty replaced by real-analytic, then P is said to be ''analytically hypoelliptic''. Every elliptic operator with C^\infty coefficients is hypoelliptic. In particular, the Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ... is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). In addition, the operator for the heat equation (P(u)=u_t - k\,\Delta u\,) :P= \partial_t - k\,\Delta_x\, (where k>0) is hypoelliptic but not elliptic. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations. Definitions Let L be a linear differential operator of order ''m'' on a domain \Omega in R''n'' given by Lu = \sum_ a_\alpha(x)\partial^\alpha u where \alpha = (\alpha_1, \dots, \alpha_n) denotes a multi-index, and \partial^\alpha u = \partial^_1 \cdots \partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atlas (topology)
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles. Charts The definition of an atlas depends on the notion of a ''chart''. A chart for a topological space ''M'' is a homeomorphism \varphi from an open subset ''U'' of ''M'' to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair (U, \varphi). When a coordinate system is chosen in the Euclidean space, this defines coordinates on U: the coordinates of a point P of U are defined as the coordinates of \varphi(P). The pair formed by a chart and such a coordinate system is called a local coordinate system, coordinate chart, coordinate patch, coordinate map, or local frame. Formal definition of at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hölder Space
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Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |