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Journal articles on the topic 'Manifolds (Mathematics) Lie groups'

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Published: 4 June 2021

Last updated: 2 February 2022

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1

Herz, Carl. "Representations of Lie Groups By Contact Transformations, I: Compact Groups." Canadian Mathematical Bulletin 33, no. 4 (1990): 369–75. http://dx.doi.org/10.4153/cmb-1990-061-6.

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2

Winkelmann, J�rg. "Realizing connected Lie groups as automorphism groups of complex manifolds." Commentarii Mathematici Helvetici 79, no. 2 (2004): 285–99. http://dx.doi.org/10.1007/s00014-003-0794-5.

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3

Alexopoulos, Georgios. "Oscillating multipliers on Lie groups and Riemannian manifolds." Tohoku Mathematical Journal 46, no. 4 (1994): 457–68. http://dx.doi.org/10.2748/tmj/1178225675.

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4

Coulhon, Thierry, Emmanuel Russ, and Valerie Tardivel-Nachef. "Sobolev algebras on Lie groups and Riemannian manifolds." American Journal of Mathematics 123, no. 2 (2001): 283–342. http://dx.doi.org/10.1353/ajm.2001.0009.

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5

Arvanitoyeorgos, Andreas, V. V. Dzhepko, and Yu G. Nikonorov. "Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups." Canadian Journal of Mathematics 61, no. 6 (2009): 1201–13. http://dx.doi.org/10.4153/cjm-2009-056-2.

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Abstract A Riemannian manifold (M, ρ) is called Einstein if the metric ρ satisfies the condition Ric(ρ) = c · ρ for some constant c. This paper is devoted to the investigation of G-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces G/H of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds SO(n)/SO(l). Furthermore, we show that for any positive integer p there exists a Stiefelmanifold SO(n)/SO(l) that admits at least p SO(n)-invariant Einstein metrics.

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6

Freibert, Marco, Lothar Schiemanowski, and Hartmut Weiss. "Homogeneous Spinor Flow." Quarterly Journal of Mathematics 71, no. 1 (2019): 21–51. http://dx.doi.org/10.1093/qmathj/haz036.

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Abstract We study the spinor flow on homogeneous spin manifolds. After providing the general setup we discuss the homogeneous spinor flow in dimension three and on almost abelian Lie groups in detail. As a further example, the flag manifold in dimension six is treated.

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7

Ivanova, Miroslava, and Lilko Dospatliev. "Geometric characteristics and properties of a three-parametric family of Lie groups with almost contact B-metric structure of the smallest dimension." Asian-European Journal of Mathematics 13, no. 08 (2020): 2050163. http://dx.doi.org/10.1142/s1793557120501636.

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Almost contact [Formula: see text]-metric manifolds of the lowest dimension 3 are constructed by a three-parametric family of Lie groups. Our aim is to determine the class of considered manifolds in a classification of almost contact [Formula: see text]-metric manifolds and their most important geometric characteristics and properties. Also, the type of the constructed Lie algebras is established in the Bianchi classification.

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8

Nikolayevsky, Y. "Weyl homogeneous manifolds modelled on compact Lie groups." Differential Geometry and its Applications 28, no. 6 (2010): 689–96. http://dx.doi.org/10.1016/j.difgeo.2010.08.002.

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9

HOFMANN, K. H., and K. H. NEEB. "Pro-Lie groups which are infinite-dimensional Lie groups." Mathematical Proceedings of the Cambridge Philosophical Society 146, no. 2 (2009): 351–78. http://dx.doi.org/10.1017/s030500410800128x.

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AbstractA pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically

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10

Suciu, Alexander I., and He Wang. "Formality properties of finitely generated groups and Lie algebras." Forum Mathematicum 31, no. 4 (2019): 867–905. http://dx.doi.org/10.1515/forum-2018-0098.

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Abstract We explore the graded-formality and filtered-formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model

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11

Applebaum, David, and Hiroshi Kunita. "Lévy flows on manifolds and Lévy processes on Lie groups." Journal of Mathematics of Kyoto University 33, no. 4 (1993): 1103–23. http://dx.doi.org/10.1215/kjm/1250519132.

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12

Oğuz, Gülay, Ilhan Içen, and Gürsoy Habil. "Lie rough groups." Filomat 32, no. 16 (2018): 5735–41. http://dx.doi.org/10.2298/fil1816735o.

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This paper introduces the definition of a Lie rough group as a natural development of the concepts of a smooth manifold and a rough group on an approximation space. Furthermore, the properties of Lie rough groups are discussed. It is shown that every Lie rough group is a topological rough group, and that the product of two Lie rough groups is again a Lie rough group. We define the concepts of Lie rough subgroups and Lie rough normal subgroups. Finally, our aim is to give an example by using definition of Lie rough homomorphism sets G and H.

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13

Gordon, Carolyn S. "Naturally Reductive Homogeneous Riemannian Manifolds." Canadian Journal of Mathematics 37, no. 3 (1985): 467–87. http://dx.doi.org/10.4153/cjm-1985-028-2.

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The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are(i) to study all naturally reductive homogeneous spaces of G when G is either semisimp

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14

Larotonda, Gabriel. "Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces." Forum Mathematicum 31, no. 6 (2019): 1567–605. http://dx.doi.org/10.1515/forum-2019-0127.

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AbstractWe study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient {M\simeq G/K}. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths.

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15

KOBAK, PIOTR Z., and ANDREW SWANN. "CLASSICAL NILPOTENT ORBITS AS HYPERKÄHLER QUOTIENTS." International Journal of Mathematics 07, no. 02 (1996): 193–210. http://dx.doi.org/10.1142/s0129167x96000116.

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We show that on an arbitrary nilpotent orbit [Formula: see text] in [Formula: see text] where [Formula: see text] is a direct sum of classical simple Lie algebras, there is a G-invariant hyperKähler structure obtainable as a hyperKäher quotient of the flat hyperKähler manifold ℝ4N≅ℍN. Coïncidences between various low-dimensional simple Lie groups lead to some nilpotent orbits being described as hyperKähler quotients (in some cases in fact finite quotients) of other nilpotent orbits. For example, from the construction we are able to read off pairs of orbits [Formula: see text] in different clas

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16

LEE, KYUNG BAI, and FRANK RAYMOND. "MAXIMAL TORUS ACTIONS ON SOLVMANIFOLDS AND DOUBLE COSET SPACES." International Journal of Mathematics 02, no. 01 (1991): 67–76. http://dx.doi.org/10.1142/s0129167x91000065.

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Any compact, connected Lie group which acts effectively on a closed aspherical manifold is a torus Tk with k ≤ rank of [Formula: see text], the center of π1 (M). When [Formula: see text], the torus action is called a maximal torus action. The authors have previously shown that many closed aspherical manifolds admit maximal torus actions. In this paper, a smooth maximal torus action is constructed on each solvmanifold. They also construct smooth maximal torus actions on some double coset spaces of general Lie groups as applications.

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17

Ludkovsky, S. V. "Stochastic processes on totally disconnected topological groups." International Journal of Mathematics and Mathematical Sciences 2003, no. 48 (2003): 3067–89. http://dx.doi.org/10.1155/s0161171203108162.

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Stochastic processes on totally disconnected topological groups are investigated. In particular, they are considered for diffeomorphism groups and loop groups of manifolds on non-Archimedean Banach spaces. Theorems about a quasi-invariance and a pseudodifferentiability of transition measures are proved. Transition measures are used for the construction of strongly continuous representations including the irreducible ones of these groups. In addition, stochastic processes on general Banach-Lie groups, loop monoids, loop spaces, and path spaces of manifolds on Banach spaces over non-Archimedean

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18

IANNUZZI, ANDREA, ANDREA SPIRO, and STEFANO TRAPANI. "COMPLEXIFICATIONS OF HOLOMORPHIC ACTIONS AND THE BERGMAN METRIC." International Journal of Mathematics 15, no. 08 (2004): 735–47. http://dx.doi.org/10.1142/s0129167x04002533.

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Let G=(ℝ,+) act by biholomorphisms on a Stein manifold X which admits the Bergman metric. We show that X can be regarded as a G-invariant domain in a "universal" complex manifold X* on which the complexification [Formula: see text] of G acts. The analogous result holds for actions of a larger class of real Lie groups containing, e.g. abelian and certain nilpotent ones. For holomorphic actions of such groups on Stein manifolds, necessary and sufficient conditions for the existence of X* are given.

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19

Fegan, H. D., and B. Steer. "First Order Operators on Manifolds With a Group Action." Canadian Journal of Mathematics 48, no. 4 (1996): 758–76. http://dx.doi.org/10.4153/cjm-1996-039-6.

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AbstractWe investigate questions of spectral symmetry for certain first order differential operators acting on sections of bundles over manifolds which have a group action. We show that if the manifold is in fact a group we have simple spectral symmetry for all homogeneous operators. Furthermore if the manifold is not necessarily a group but has a compact Lie group of rank 2 or greater acting on it by isometries with discrete isotropy groups, and let D be a split invariant elliptic first order differential operator, then D has equivariant spectral symmetry.

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20

Aloui, Foued, and Nadhem Zaalani. "Reduced Riemannian Poisson manifolds and Riemannian Poisson-Lie groups." Differential Geometry and its Applications 68 (February 2020): 101582. http://dx.doi.org/10.1016/j.difgeo.2019.101582.

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21

Zhou, Jianwei. "Morse functions on Grassmann manifolds." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135, no. 1 (2005): 209–21. http://dx.doi.org/10.1017/s030821050000384x.

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In this paper we study the Morse functions on Grassmann manifolds. It is shown that many well-known calibrations can be used to construct degenerate Morse functions on Grassmann manifolds. In the study we use Clifford algebras and Lie groups.

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22

YOKOYAMA, TOMOO. "Fixed points of certain Anosov maps on Riemannian manifolds." Ergodic Theory and Dynamical Systems 30, no. 4 (2009): 1261–70. http://dx.doi.org/10.1017/s0143385709000492.

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23

Gualtieri, Marco, Mykola Matviichuk, and Geoffrey Scott. "Deformation of Dirac Structures via L∞ Algebras." International Mathematics Research Notices 2020, no. 14 (2018): 4295–323. http://dx.doi.org/10.1093/imrn/rny134.

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Abstract The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra that depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty $ algebra instead. We develop a simplified method for describing this $L_\infty $ algebra and use it to prove that the $L_\infty $ algebras corresponding to different transversals are canonically $L_\infty $–isomorphic. In some cases, this isomorphism provides a formality map, as we show in several examples including (quasi)-Poisson geometry, Dirac structures

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24

Berenstein, Carlos A. "INVARIANT FUNCTION SPACES ON HOMOGENEOUS MANIFOLDS OF LIE GROUPS AND APPLICATIONS." Bulletin of the London Mathematical Society 28, no. 1 (1996): 99–102. http://dx.doi.org/10.1112/blms/28.1.99.

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25

Bernhard Lamel, Nordine Mir, and Dmitri Zaitsev. "Lie group structures on automorphism groups of real-analytic CR manifolds." American Journal of Mathematics 130, no. 6 (2008): 1709–26. http://dx.doi.org/10.1353/ajm.0.0033.

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26

Huang, Yonghong, and Shanzhong Sun. "Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds." Frontiers of Mathematics in China 15, no. 1 (2020): 91–114. http://dx.doi.org/10.1007/s11464-020-0823-3.

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27

Papadima, Stefan, and Alexander I. Suciu. "The Topology of Compact Lie Group Actions Through the Lens of Finite Models." International Mathematics Research Notices 2019, no. 20 (2018): 6390–436. http://dx.doi.org/10.1093/imrn/rnx294.

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AbstractGiven a compact, connected Lie group K, we use principal K-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let M be a compact, connected, smooth manifold, which supports an almost free K-action. Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the action, we describe an algebraic model for M with commensurate finiteness and partial formality properties. The existence of such a model has various implications on the structure of the cohomology jump loci of M and of the repr

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28

Hirica, Iulia-Elena, Cristina-Liliana Pripoae, Gabriel-Teodor Pripoae, and Vasile Preda. "Affine Differential Geometric Control Tools for Statistical Manifolds." Mathematics 9, no. 14 (2021): 1654. http://dx.doi.org/10.3390/math9141654.

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The paper generalizes and extends the notions of dual connections and of statistical manifold, with and without torsion. Links with the deformation algebras and with the Riemannian Rinehart algebras are established. The semi-Riemannian manifolds admitting flat dual connections with torsion are characterized, thus solving a problem suggested in 2000 by S. Amari and H. Nagaoka. New examples of statistical manifolds are constructed, within and beyond the classical setting. The invariant statistical structures on Lie groups are characterized and the dimension of their set is determined. Examples f

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29

Rosales-Ortega, José. "The signature in actions of semisimple Lie groups on pseudo-Riemannian manifolds." Proyecciones (Antofagasta) 31, no. 1 (2012): 51–63. http://dx.doi.org/10.4067/s0716-09172012000100006.

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30

Liao, Ming. "Stable Manifolds for Stochastic Flows Induced by Lévy Processes on Lie Groups." Proceedings of the London Mathematical Society 83, no. 2 (2001): 493–512. http://dx.doi.org/10.1112/s002461150101303x.

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31

Dungey, Nick. "Some remarks on gradient estimates for heat kernels." Abstract and Applied Analysis 2006 (2006): 1–10. http://dx.doi.org/10.1155/aaa/2006/73020.

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This paper is concerned with pointwise estimates for the gradient of the heat kernelKt,t>0, of the Laplace operator on a Riemannian manifoldM. Under standard assumptions onM, we show that∇Ktsatisfies Gaussian bounds if and only if it satisfies certain uniform estimates or estimates inLpfor some1≤p≤∞. The proof is based on finite speed propagation for the wave equation, and extends to a more general setting. We also prove that Gaussian bounds on∇Ktare stable under surjective, submersive mappings between manifolds which preserve the Laplacians. As applications, we obtain gradient estimates on

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32

Brunsden, Victor. "Local rigidity and group cohomology I: Stowe's theorem for Banach manifolds." Bulletin of the Australian Mathematical Society 59, no. 2 (1999): 271–95. http://dx.doi.org/10.1017/s0004972700032895.

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Stowe's Theorem on the stability of the fixed points of a C2 action of a finitely generated group Γ is generalised to C1 actions of such groups on Banach manifolds. The result is then used to prove that if φ is a Cr action on a smooth, closed, manifold M satisfying H1(Γ, Dr−1(M)) = 0, then φ is locally rigid. Here, r ≥ 2 and Dk(M) is the space of Ck tangent vector fields on M. This generalises a local rigidity result of Weil for representations of a finitely generated group Γ in a Lie group.

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33

Chon, I., and J. D. Lawson. "Attainable sets and one-parameter semigroups of sets." Glasgow Mathematical Journal 33, no. 2 (1991): 187–201. http://dx.doi.org/10.1017/s0017089500008223.

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The methods of Lie theory have found widespread application in the study of the Lie algebras of vector fields on manifolds that arise naturally in geometric control theory (for some such applications, see [1]). Control systems on Lie groups themselves also have received considerable attention (see, for example, [9]). After reviewing basic facts about control systems on Lie groups, we derive the close relationship between attainable sets and Rådström's theory [12] of one-parameter semigroups of sets (Section 2). These ideas are then linked to the recently emerging Lie theory of semigroups [5].

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34

Hillman, Jonathan A. "Deficiencies of lattices in connected Lie groups." Bulletin of the Australian Mathematical Society 65, no. 3 (2002): 393–97. http://dx.doi.org/10.1017/s0004972700020438.

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We complete the determination of the groups of positive deficiency which occur as lattices in connected Lie groups. The torsion free groups among them are 3-mainfold groups. We show that any other torsion free 3-manifold group which is such a lattice is the group of an aspherical closed geometric 3-manifold.

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35

FIORI, SIMONE. "UNSUPERVISED NEURAL LEARNING ON LIE GROUP." International Journal of Neural Systems 12, no. 03n04 (2002): 219–46. http://dx.doi.org/10.1142/s012906570200114x.

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The present paper aims at introducing the concepts and mathematical details of unsupervised neural learning with orthonormality constrains. The neural structures considered are single non-linear layers and the learnable parameters are organized in matrices, as usual, which gives the parameters spaces the geometrical structure of the Euclidean manifold. The constraint of orthonormality for the connection-matrices further restricts the parameters spaces to differential manifolds such as the orthogonal group, the compact Stiefel manifold and its extensions. For these reasons, the instruments for

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36

PEYERIMHOFF, NORBERT, and EVANGELIA SAMIOU. "A support theorem for the X-ray transform on manifolds with plane covers." Mathematical Proceedings of the Cambridge Philosophical Society 169, no. 1 (2019): 149–58. http://dx.doi.org/10.1017/s0305004119000148.

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AbstractThis paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan. We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.

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37

Kim, Sung-Yeon. "Analytic Jet Parametrization for CR Automorphisms of Some Essentially Finite CR Manifolds." Nagoya Mathematical Journal 189 (2008): 155–68. http://dx.doi.org/10.1017/s0027763000009533.

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AbstractIn this paper we construct analytic jet parametrizations for the germs of real analytic CR automorphisms of some essentially finite CR manifolds on their finite jet at a point. As an application we show that the stability groups of such CR manifolds have Lie group structure under composition with the topology induced by uniform convergence on compacta.

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38

Dancer, Andrew, and Andrew Swann. "Hyperkähler metrics associated to compact Lie groups." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 1 (1996): 61–69. http://dx.doi.org/10.1017/s0305004100074673.

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It is well known that the cotangent bundle of any manifold has a canonical symplectic structure. If we specialize to the case when the manifold is a compact Lie group G, then this structure is preserved by the actions of G on T*G induced by left and right translation on G. We refer to these as the left and right actions of G on T*G.

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39

Russ, Emmanuel, and Yannick Sire. "Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds." Studia Mathematica 203, no. 2 (2011): 105–27. http://dx.doi.org/10.4064/sm203-2-1.

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40

Huang, Jing-Song, Yongzhi Luan, and Binyong Sun. "On Borel–de Siebenthal Representations." International Mathematics Research Notices 2019, no. 15 (2018): 4845–58. http://dx.doi.org/10.1093/imrn/rnx304.

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AbstractHolomorphic representations are lowest weight representations for simple Lie groups of Hermitian type and have been studied extensively. Inspired by the work of Kobayashi on discrete series for indefinite Stiefel manifolds, Gross–Wallach on quaternonic discrete series and their analytic continuation, and Ørsted–Wolf on Borel–de Siebenthal discrete series, we define and study Borel–de Siebenthal representations (also called quasi-holomorphic representations) associated with Borel–de Siebenthal root systems for simple Lie groups of non-Hermitian type.

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41

Mantegazza, Mauro. "Construction of projective special Kähler manifolds." Annali di Matematica Pura ed Applicata (1923 -) 200, no. 6 (2021): 2645–87. http://dx.doi.org/10.1007/s10231-021-01096-4.

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AbstractIn this paper, we present an intrinsic characterisation of projective special Kähler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special Kähler structure on $$\mathrm {SU}(n,1)/\mathrm {S}(\mathrm {U}(n)\mathrm {U}(1))$$ SU ( n , 1 ) / S ( U ( n ) U ( 1 ) ) . We use this characterisation to classify 4-dimensional projective special Kähler Lie groups.

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42

Smoczyk, Knut. "Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature." Calculus of Variations and Partial Differential Equations 4, no. 2 (1996): 155–70. http://dx.doi.org/10.1007/bf01189952.

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43

Smoczyk, Knut. "Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature." Calculus of Variations and Partial Differential Equations 4, no. 2 (1996): 155–70. http://dx.doi.org/10.1007/s005260050033.

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44

Kruglikov, Boris, and Henrik Winther. "Submaximally symmetric quaternion Hermitian structures." International Journal of Mathematics 31, no. 11 (2020): 2050084. http://dx.doi.org/10.1142/s0129167x20500846.

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We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric properties of the submaximally symmetric spaces are studied, in particular, we identify locally conformally quaternion-Kähler structures as well as quaternion-Kähler with torsion.

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45

Wang, Wenjie, and Xinxin Dai. "Pseudo-Parallel Characteristic Jacobi Operators on Contact Metric 3 Manifolds." Journal of Mathematics 2021 (July 21, 2021): 1–6. http://dx.doi.org/10.1155/2021/6148940.

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We prove that the characteristic Jacobi operator on a contact metric three manifold is semiparallel if and only if it vanishes. We determine Lie groups of dimension three admitting left invariant contact metric structures such that the characteristic Jacobi operators are pseudoparallel.

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46

Owczarek, Robert. "Spinor structures on flag manifolds of compact simple Lie groups." Journal of Physics A: Mathematical and General 32, no. 21 (1999): 3979–91. http://dx.doi.org/10.1088/0305-4470/32/21/310.

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47

Janssens, Bas, and Christoph Wockel. "Universal central extensions of gauge algebras and groups." crll 2013, no. 682 (2012): 129–39. http://dx.doi.org/10.1515/crelle-2012-0021.

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Abstract. We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed by Neeb and Wockel (2009), is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting.

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48

Herz, Carl. "Representations of Lie Groups by Contact Transformations, II: Non-Compact Simple Groups." Canadian Journal of Mathematics 45, no. 4 (1993): 778–802. http://dx.doi.org/10.4153/cjm-1993-044-x.

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AbstractIf a Lie group acts faithfully as a transitive group of contact transformations of a compact manifold it is either compact with centre of dimension at most 1 or non-compact simple. The latter case is described

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49

Zolotov, Vladimir. "FINITE FLAT SPACES." Mathematika 65, no. 4 (2019): 1010–17. http://dx.doi.org/10.1112/s0025579319000263.

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We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$ if for every $\unicode[STIX]{x1D716}>0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less than $1+\unicode[STIX]{x1D716}$. We show that almost isometric embeddability conditions are equal for the following classes of spaces.(a)Quotients of Euclidean spaces by isometric actions of finite groups.(b)$L_{2}$-Wasserstein spaces over Euclidean spaces.(c)Compact flat manifolds.(d)Compact flat orbifolds.(e)Quotients of connected compact bi

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50

KAWAKUBO, Katsuo. "$G$ - $s$ -cobordant manifolds are not necessarily $G$ -homeomorphic for arbitrary compact Lie groups $G$." Journal of the Mathematical Society of Japan 45, no. 4 (1993): 599–610. http://dx.doi.org/10.2969/jmsj/04540599.

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