Locally Homogeneous $$C^0$$ -Riemannian Manifolds

Transformation Groups https://doi.org/10.1007/s00031-023-09811-6 Transformation Groups Locally Homogeneous C 0 -Riemannian Manifolds Nina Lebedeva1,2 · Artem Nepechiy3 Received: 28 April 2022 / Accepted: 27 April 2023 © The Author(s) 2023 Abstract We show that locally homogeneous C 0 -Riemannian manifolds are smooth. Keywords Riemannian manifolds of low regularity · Locally homogeneous spaces Mathematics Subject Classification (2010) 53C30 · 57S05 · 58D05 1 Introduction In this paper we prove that if a C 0 -Riemannian manifold is locally homogeneous, then it is indeed smooth, more precisely we obtain the following theorem: Main Theorem (Local homogeneity implies smoothness). Let (M, g0 ) be a locally homogeneous C 0 -Riemannian manifold and denote by dg0 the induced metric, then (M, dg0 ) is isometric to a smooth Riemannian manifold. In fact we show that for any point there is a small neighborhood U , such that the set of local isometries on U , which will be denoted by UG , forms a local Lie group with Lie algebra g acting transitively on U . The isotropy local isometries determine a local Lie group U H with Lie algebra h and U is isometric to the coset space UG /U H carrying an invariant metric with respect to the left action of UG (for definitions see [20, 23, 29, 30]). In particular all spaces appearing in the main theorem are determined by Lie algebras g ⊃ h together with a scalar product ·, · on g/h, which is skew symmetric with respect to the adjoint action of h on g/h [30]. Thus, they are given by B Artem Nepechiy Nina Lebedeva 1 Saint Petersburg State University, 7/9 Universitetskaya nab, St. Petersburg 199034, Russia 2 St. Petersburg Department of V.A., Steklov Institute of Mathematics of the Russian Academy of Sciences, 27 Fontanka nab, 191023 St. Petersburg, Russia 3 Institute for Algebra and Geometry, KIT, Englerstr. 2, 76131 Karlsruhe, Germany N. Lebedeva and A. Nepechiy purely algebraic data. Moreover, this implies that M and its Riemannian metric are real analytic. Our result in some sense generalizes the Myers-Steenrod theorems [22], which in particular assert that the isometry group of a smooth Riemannian manifold is a Lie group. Several theorems are known in that direction: Metric spaces with geometric assumptions such as curvature conditions imply regularity of the isometry group. For example isometry groups of Alexandrov spaces or RCD*(K , N ) spaces are known to be Lie groups [10, 13, 33]. The question "When is a homogeneous/locally homogeneous space a smooth manifold?" has been investigated in [1, 25, 26]. In [1][Theorem 7] Berestovskii studied when a globally homogeneous inner metric space is isometric to a homogeneous Riemannian manifold. His findings show in particular that a homogeneous Alexandrov space is in fact a smooth Riemannian manifold. In contrast to that we obtain a theorem of a local nature. One can show (using [17, 18] for upper curvature bounds and [24] for lower bounds) that a locally homogeneous space with an upper or lower curvature bound in the sense of Alexandrov is a C 0 -Riemannian manifold. Hence our main theorem implies: Corollary Let X be a locally homogeneous, locally compact, length space of finite Hausdorff dimension. If there exists a point together with a convex neighborhood admitting a curvature bound from either above or below in the sense of Alexandrov, then X is isometric to a smooth Riemannian manifold. It would be interesting to obtain a full description of locally homogeneous, locally compact length spaces similar to [1] without assuming any regularity on the metric and topology. This could be considered as a metric version of the Bing-Borsuk conjecture [14]. There exist different results in the local setting; however, they are making stronger assumptions on the regularity of the manifold. In [28] Singer showed: If a complete, simply connected Riemannian manifold is curvature homogeneous and the derivatives of the curvature tensor agree up to some order at all points, then the manifold is globally homogeneous. If the Riemannian metric is complete and sufficiently smooth, the conclusion of our main theorem follows from this result. While the proof is essentially local and completeness does not play a central role, it relies heavily on the existence of high order derivatives of the metric [23, 28]. Lately local versions with lower regularity have been obtained by Pediconi [25, 26] with different additional assumptions on the space and the group action. Riemannian manifolds with low regularity do not satisfy classical results in Riemannian geometry: There is no meaningful notion of curvature and shortest curves do not need to solve a differential equation, they may branch and the injectivity radius may be zero [15]. Shortest curves do not even need to be C 1 [15]. We refer to [4] for some basic properties of C 0 -Riemannian manifolds and to [5, 7, 8] for further results. A metric space M is called locally homogeneous if for any two points of M there is a local isometry taking one to the other. One important problem and the difference to the non-local case (as considered by Berestovskii) is that the set of local isometries is a priori not known to form a local group. The technical tool to overcome this obstacle is to extend local isometries, defined on arbitrary small balls, to balls of fixed radius. Locally Homogeneous Once a local group structure is established one can apply structure theory of locally compact groups [12, 27] to show that it is a local Lie group. We then construct a local isometry between our metric space M and a local quotient of the local group equipped with an invariant Riemannian metric. The paper is organized as follows: In Section 2 we fix notation, explain what a C 0 -Riemannian manifold is and give definitions and notions regarding local groups. In Section 3 we prove that every local isometry can be extended to an isometry of fixed size. In Section 4 we explain how to obtain a local topological group and prove that some restriction is a local Lie group. After that we will explain how to obtain a left-invariant metric on the quotient, which is isometric to some open subset of M. 2 Preliminaries 2.1 C0 -Riemannian Manifolds In this subsection we collect all definitions and results regarding C 0 -Riemannian manifolds. Definition 2.1 (C 0 -Riemannian manifold) A C 0 -Riemannian manifold is a pair (M, g0 ) consisting of a C 1 -manifold M together with a continuous Riemannian metric g0 . The Riemannian metric g0 induces a canonical length structure, which in turn induces an intrinsic metric dg0 on M. This allows us to formulate local homogeneity in purely metric terms. We denote open (closed) balls with radius r around the point x by Br (x) (B r (x)). Definition 2.2 (Local homogeneity) A metric space M is called locally homogeneous if for every x, y ∈ M there exists r > 0 and an isometry f : Br (x) → Br (y) satisfying f (x) = y. We call such a map f a pointed isometry. (...truncated)