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The moduli space of locally homogeneous spaces and
locally homogeneous spaces which are not locally isometric to globally
homogeneous spaces

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*Kazumi Tsukada*

**E-mail:** tsukada@math.ocha.ac.jp
**Abstract.** Contrary to a natural expectation, there exist locally
homogeneous spaces which are not locally isometric to any (globally) homogeneous
spaces. Such examples were found by Kowalski. In this paper we try to understand
these mysterious examples well. We denote by $\Cal {LH}(n)$ the set of
local isometry classes of $n$-dimensional locally homogeneous spaces. We
can introduce the topology on $\Cal {LH}(n)$ by imbedding it in the space
of abstract curvature tensors and covariant derivatives. We show that the
set of local isometry classes of $n$-dimensional locally homogeneous spaces
which are locally isometric to homogeneous spaces are dense in $\Cal {LH}(n)$.

**AMSclassification.** 53C30, 53B20

**Keywords.** Locally homogeneous spaces, infinitesimally homogeneous
spaces