Not all vector spaces with complete translation-invariant metrics are Fr�chet spaces.
2.
Fr�chet spaces are locally convex spaces that are complete with respect to a translation invariant metric.
3.
The space for is an F-space : it admits a complete translation-invariant metric with respect to which the vector space operations are continuous.
4.
A "'compact Riemannian nilmanifold "'is a compact Riemannian manifold which is locally isometric to a nilpotent Lie group with left-invariant metric.
5.
More precisely, seven of the 8 geometries can be realized as a left-invariant metric on the simply connected group ( sometimes in more than one way ).
6.
Every locally compact group which is second-countable is metrizable as a topological group ( i . e . can be given a left-invariant metric compatible with the topology ) and complete.
7.
The requirement that the transitive nilpotent group acts by isometries leads to the following rigid characterization : every homogeneous nilmanifold is isometric to a nilpotent Lie group with left-invariant metric ( see Wilson ).
8.
This distance provides a right-invariant metric of diffeomorphometry, invariant to reparameterization of space since for all \ varphi \ in \ operatorname { Diff } _ V \, \, \,,
9.
This geometry can be modeled as a left invariant metric on the Bianchi group of type II . Finite volume manifolds with this geometry are compact and orientable and have the structure of a Seifert fiber space.
10.
If the locally compact abelian group " G " is separable and metrizable ( its topology may be defined by a translation-invariant metric ) then harmonious sets admit another, related, description.
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