IsotropicSpace - crowlogic/arb4j GitHub Wiki

Isotropic Spaces

Definition: An isotropic space is a space in which all directions are equivalent in terms of properties and structure. More formally, a space $X$ is isotropic if the group of isometries $G$ acting on $X$ acts transitively on the unit sphere in the tangent space $T_pX$ at any point $p \in X$.

Mathematical Formulation:

1. Riemannian Manifold: Let $(X, g)$ be a Riemannian manifold, where $X$ is a set and $g$ is a metric tensor.
2. Isometry Group: The group of isometries $G$ of $X$ preserves the metric tensor $g$.
3. Transitive Action: For any point $p \in X$, the action of $G$ on the unit sphere in $T_pX$ (the tangent space at $p$) is transitive.

Semi-Isotropic Spaces

Definition: A semi-isotropic space is a space where isotropy (equivalence of all directions) is present in some, but not all, directions or dimensions.

Mathematical Formulation:

1. Riemannian Manifold: Again, let $(X, g)$ be a Riemannian manifold.
2. Partial Isometry Group: Let $H \subset G$ be a subgroup of the full isometry group $G$ of $X$.
3. Partial Transitive Action: For any point $p \in X$, the action of $H$ is transitive on a subset of the unit sphere in $T_pX$, but not necessarily on the entire sphere.