Isotropic manifold

Encyclopedia

In mathematics

, an

in which the geometry

doesn't depend on directions. A simple example is the surface of a sphere.

A homogeneous space

is a similar concept. A homogeneous space can be non-isotropic (for example, a flat torus), in the sense that an invariant metric tensor

on a homogeneous space may not be isotropic.

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an

**isotropic manifold**is a manifoldManifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

in which the geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

doesn't depend on directions. A simple example is the surface of a sphere.

A homogeneous space

Homogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...

is a similar concept. A homogeneous space can be non-isotropic (for example, a flat torus), in the sense that an invariant metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

on a homogeneous space may not be isotropic.