Manifold in which the geometry does not depend on directions
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Not to be confused with isotropic subspace, a quadratic space containing a non-zero vector v for which q(v) is 0.
In mathematics, an isotropic manifold is a manifold in which the geometry does not depend on directions. Formally, we say that a Riemannian manifold is isotropic if for any point and unit vectors , there is an isometry of with and . Every connected isotropic manifold is homogeneous, i.e. for any there is an isometry of with This can be seen by considering a geodesic from to and taking the isometry which fixes and maps to
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mathematics, an isotropicmanifold is a manifold in which the geometry does not depend on directions. Formally, we say that a Riemannian manifold ( M , g )...
of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is co-isotropic (the dual of an isotropic subspace), the...
an isotropic vector is an isotropic line. Isotropic coordinates Isotropic coordinates are coordinates on an isotropic chart for Lorentzian manifolds. Isotropy...
This means that (except in the trivial case of a locally flat manifold), the angular isotropic coordinates do not faithfully represent distances within the...
on an infinite dimensional maximal isotropic Grassmann manifold. Finite dimensional positive Grassmann manifolds can be used to express soliton solutions...
almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost...
not positivity. For example, a manifold with an inner product structure on its tangent spaces is a Riemannian manifold, while relaxing this to a symmetric...
geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure...
being isotropic, that is, there is no property of the space that allows distinguishing between two points or two lines. Therefore, a more isotropic definition...
approach to spacetime manifolds. In the Verma module of a Lie algebra there are null vectors. Emil Artin (1957) Geometric Algebra, isotropic Arthur A. Sagle...
an n-manifold, considered as a 2n-manifold, has an almost complex structure, and the cotangent bundle of an n-manifold, considered as a 2n-manifold, has...
a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties...
locally from flat space. The curvature of any locally isotropic space (and hence of a locally isotropic universe) falls into one of the three following cases:...
Riemannian manifolds Intrinsic metric – Concept in geometry/topology Isotropic line Jacobi field Morse theory – Analyzes the topology of a manifold by studying...
which are homogeneous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic. (This list is closely related...
the radiation pattern of non-isotropic antenna elements, which effects main and grating lobes differently. For isotropic antenna elements, the main and...
simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches...
determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and...
(1647–1748) before Euler's formulation with e Hyperbolic manifold, a complete Riemannian n-manifold of constant sectional curvature −1 Hyperbolic motion,...
annihilated, under the Clifford algebra representation, by a maximal isotropic subspace of a vector space V {\displaystyle V} with respect to a scalar...
and isotropic medium, with u(x, y, z, t) being the temperature at the point (x, y, z) and time t. If the medium is not homogeneous and isotropic, then...
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...
physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They...
Michelson–Morley experiment is that the round-trip travel time for light is isotropic (independent of direction), but the result alone is not enough to discount...
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