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Journal articles on the topic 'Invariant Riemannian metrics'

Author: Grafiati

Published: 10 March 2023

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1

Wang, Hui, and Shaoqiang Deng. "Left Invariant Einstein–Randers Metrics on Compact Lie Groups." Canadian Mathematical Bulletin 55, no. 4 (December 1, 2012): 870–81. http://dx.doi.org/10.4153/cmb-2011-145-6.

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AbstractIn this paper we study left invariant Einstein–Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein–Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein–Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

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2

Parhizkar, M., and D. Latifi. "On the flag curvature of invariant (α,β)-metrics." International Journal of Geometric Methods in Modern Physics 13, no. 04 (March 31, 2016): 1650039. http://dx.doi.org/10.1142/s0219887816500390.

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In this paper, we consider invariant [Formula: see text]-metrics which are induced by invariant Riemannian metrics [Formula: see text] and invariant vector fields [Formula: see text] on homogeneous spaces. We study the flag curvatures of invariant [Formula: see text]-metrics. We first give an explicit formula for the flag curvature of invariant [Formula: see text]-metrics arising from invariant Riemannian metrics on homogeneous spaces and Lie groups. We then give some explicit formula for the flag curvature of invariant Matsumoto metrics, invariant Kropina metrics and invariant Randers metrics.

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3

Balashchenko, V. V., P. N. Klepikov, E. D. Rodionov, and O. P. Khromova. "On the Cerbo Conjecture on Lie Groups with the Left-Invariant Lorentzian Metric." Izvestiya of Altai State University, no. 1(123) (March 18, 2022): 79–82. http://dx.doi.org/10.14258/izvasu(2022)1-12.

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Manifolds with constraints on tensor fields include Einstein manifolds, Einstein-like manifolds, conformally flat manifolds, and a number of other important classes of manifolds. The work of many mathematicians is devoted to the study of such manifolds, which is reflected in the monographs of A. Besse, M. Berger, M.-D. Cao, M. Wang. Ricci solitons are one of the natural generalizations of Einstein's metrics. If a Riemannian manifold is a Lie group, one speaks of invariant Ricci solitons. Invariant Ricci solitons were studied in most detail in the case of unimodular Lie groups with left-invariant Riemannian metrics and the case of low dimension. Thus, L. Cerbo proved that all invariant Ricci solitons are trivial on unimodular Lie groups with left-invariant Riemannian metric and Levi-Civita connection.A similar result up to dimension four was obtained by P.N. Klepikov and D.N. Oskorbin for the non-unimodular case. We study invariant Ricci solitons on three-dimensional unimodular Lie groups with the Lorentzian metric.The study results show that unimodular Lie groups with left-invariant Lorentzian metric admit invariant Ricci solitons other than trivial ones. In this paper, a complete classification of invariant Ricci solitons on three-dimensional unimodular Lie groups with leftinvariant Lorentzian metric is obtained.

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Hashinaga, Takahiro, and Hiroshi Tamaru. "Three-dimensional solvsolitons and the minimality of the corresponding submanifolds." International Journal of Mathematics 28, no. 06 (May 2, 2017): 1750048. http://dx.doi.org/10.1142/s0129167x17500483.

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In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.

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Asgari, Farhad, and Hamid Reza Salimi Moghaddam. "Left invariant Randers metrics of Berwald type on tangent Lie groups." International Journal of Geometric Methods in Modern Physics 15, no. 01 (December 19, 2017): 1850015. http://dx.doi.org/10.1142/s0219887818500159.

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Let [Formula: see text] be a Lie group equipped with a left invariant Randers metric of Berward type [Formula: see text], with underlying left invariant Riemannian metric [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are lifted Randers and Riemannian metrics arising from [Formula: see text] and [Formula: see text] on the tangent Lie group [Formula: see text] by vertical and complete lifts. In this paper, we study the relations between the flag curvature of the Randers manifold [Formula: see text] and the sectional curvature of the Riemannian manifold [Formula: see text] when [Formula: see text] is of Berwald type. Then we give all simply connected three-dimensional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors.

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chen, Chao, Zhiqi chen, and Yuwang Hu. "Einstein metrics and Einstein–Randers metrics on a class of homogeneous manifolds." International Journal of Geometric Methods in Modern Physics 15, no. 04 (March 13, 2018): 1850052. http://dx.doi.org/10.1142/s0219887818500524.

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In this paper, we give [Formula: see text]-invariant Einstein metrics on a class of homogeneous manifolds [Formula: see text], and then prove that every homogeneous manifold [Formula: see text] admits at least three families of [Formula: see text]-invariant non-Riemannian Einstein–Randers metrics.

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7

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 (December 1, 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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8

Vylegzhanin, D. V., P. N. Klepikov, E. D. Rodionov, and O. P. Khromova. "On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton." Izvestiya of Altai State University, no. 4(120) (September 10, 2021): 86–90. http://dx.doi.org/10.14258/izvasu(2021)4-13.

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Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians. Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case. This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.

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9

Deng, Shaoqiang, and Zixin Hou. "Invariant Randers metrics on homogeneous Riemannian manifolds." Journal of Physics A: Mathematical and General 39, no. 18 (April 19, 2006): 5249–50. http://dx.doi.org/10.1088/0305-4470/39/18/c01.

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10

Deng, Shaoqiang, and Zixin Hou. "Invariant Randers metrics on homogeneous Riemannian manifolds." Journal of Physics A: Mathematical and General 37, no. 15 (March 29, 2004): 4353–60. http://dx.doi.org/10.1088/0305-4470/37/15/004.

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11

Alves, Luciana Aparecida, and Neiton Pereira da Silva. "Invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$." Boletim da Sociedade Paranaense de Matemática 38, no. 1 (February 19, 2018): 227. http://dx.doi.org/10.5269/bspm.v38i1.36604.

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It is well known that the Einstein equation on a Riemannian flag manifold $(G/K,g)$ reduces to an algebraic system if $g$ is a $G$-invariant metric. In this paper we obtain explicitly new invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$; and we compute the Einstein system for generalized flag manifolds of type $Sp(n)$. We also consider the isometric problem for these Einstein metrics.

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Hosseini, Masoumeh, and Hamid Reza Salimi Moghaddam. "Classification of Douglas (α,β)-metrics on five-dimensional nilpotent Lie groups." International Journal of Geometric Methods in Modern Physics 17, no. 08 (June 23, 2020): 2050112. http://dx.doi.org/10.1142/s0219887820501121.

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In this paper, we classify all simply connected five-dimensional nilpotent Lie groups which admit [Formula: see text]-metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During this classification, we give the geodesic vectors, Levi-Civita connection, curvature tensor, sectional curvature and [Formula: see text]-curvature.

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13

Gordon, Carolyn S. "Naturally Reductive Homogeneous Riemannian Manifolds." Canadian Journal of Mathematics 37, no. 3 (June 1, 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 semisimple of noncompact type or nilpotent and(ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

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14

Nikolayevsky, Y., and Yu G. Nikonorov. "On invariant Riemannian metrics on Ledger–Obata spaces." manuscripta mathematica 158, no. 3-4 (April 9, 2018): 353–70. http://dx.doi.org/10.1007/s00229-018-1029-9.

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15

Stoica, O. C. "On singular semi-Riemannian manifolds." International Journal of Geometric Methods in Modern Physics 11, no. 05 (May 2014): 1450041. http://dx.doi.org/10.1142/s0219887814500418.

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On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this paper, we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.

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16

APANASOV, BORIS N. "KOBAYASHI CONFORMAL METRIC ON MANIFOLDS, CHERN-SIMONS AND η-INVARIANTS." International Journal of Mathematics 02, no. 04 (August 1991): 361–82. http://dx.doi.org/10.1142/s0129167x9100020x.

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The main aim of this paper is to present a canonical Riemannian smooth metric on a given uniformized conformal manifold (conformally flat manifold) which is compatible with the conformal structure. This metric is related to the Kobayashi construction for complex-analytic manifolds and gives a new conformal invariant. As an application, the paper studies the Chern-Simons functional and the η-invariant associated with the conformal class of conformally-Euclidean metrics on a closed hyperbolic 3-manifold.

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17

Rodionov, E. D., and O. P. Khromova. "On the δ-Pinching Function of the Sectional Curvature of a Compact Connected Lie Group G with a Bi-Invariant Riemannian Metric and a Vectorial Torsion Connection." Izvestiya of Altai State University, no. 4(114) (September 9, 2020): 117–20. http://dx.doi.org/10.14258/izvasu(2020)4-19.

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One of the important problems of Riemannian geometry is the problem of establishing connections between curvature and the topology of a Riemannian manifold, and, in particular, the influence of the sign of sectional curvature on the topological structure of a Riemannian manifold. Of particular importance in these studies is the question of the influence of d-pinching of Riemannian metrics of positive sectional curvature on the geometric and topological structure of the Riemannian manifold. This question is most studied for the homogeneous Riemannian case. In this direction, the classification of homogeneous Riemannian manifolds of positive sectional curvature, obtained by M. Berger, N. Wallach, L. Bergeri, as well as a number of results on d- pinching of homogeneous Riemannian metrics of positive sectional curvature, is well known. In this paper, we investigate Riemannian manifolds with metric connection being a connection with vectorial torsion. The Levi-Civita connection falls into this class of connections. Although the curvature tensor of these connections does not possess the symmetries of the Levi-Civita connection curvature tensor, it seems possible to determine sectional curvature. This paper studies the d-pinch function of the sectional curvature of a compact connected Lie group G with a biinvariant Riemannian metric and a connection with vectorial torsion. It is proved that it takes the values d(||V ||)∈(0,1].

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18

Xu, Na, Zhiqi Chen, and Ju Tan. "Left invariant pseudo-Riemannian metrics on solvable Lie groups." Journal of Geometry and Physics 137 (March 2019): 247–54. http://dx.doi.org/10.1016/j.geomphys.2018.08.014.

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19

GOURMELON, NIKOLAZ. "Adapted metrics for dominated splittings." Ergodic Theory and Dynamical Systems 27, no. 6 (December 2007): 1839–49. http://dx.doi.org/10.1017/s0143385707000272.

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AbstractA Riemannian metric is adapted to a hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction of the unstable/stable directions is seen after only one iteration. A dominated splitting is a notion of weak hyperbolicity where the tangent bundle of the manifold splits in invariant subbundles such that the vector expansion on one bundle is uniformly smaller than that on the next bundle. The existence of an adapted metric for a dominated splitting has been considered by Hirsch, Pugh and Shub (M. Hisch, C. Pugh and M. Shub. Invariant Manifolds(Lecture Notes in Mathematics, 583). Springer, Berlin, 1977). This paper gives a complete answer to this problem, building adapted metrics for dominated splittings and partially hyperbolic splittings in arbitrarily many subbundles of arbitrary dimensions. These results stand for diffeomorphisms and for flows.

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del Barco, Viviana, and Andrei Moroianu. "Conformal Killing forms on 2-step nilpotent Riemannian Lie groups." Forum Mathematicum 33, no. 5 (July 28, 2021): 1331–47. http://dx.doi.org/10.1515/forum-2021-0026.

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Abstract We study left-invariant conformal Killing 2- or 3-forms on simply connected 2-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing 2- and 3-forms are the following: The Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing 2- and 3-forms is provided in each case.

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21

Klepikov, Pavel N., Evgeny D. Rodionov, and Olesya P. Khromova. "Mathematical modeling in the study of semisymmetric connections on three-dimensional Lie groups with the metric of the Ricci soliton." Yugra State University Bulletin 60, no. 1 (December 23, 2021): 23–29. http://dx.doi.org/10.17816/byusu20210123-29.

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Semisymmetric connections were first discovered by E. Cartan and are a natural generalization of the Levi-Civita connection. The properties of the parallel transfer of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano and other mathematicians. In this paper, a mathematical model is constructed for studying semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional unimodular Lie groups with left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that in this case there are nontrivial invariant semisimetric connections. Previously, the authors carried out similar studies in the class of Einstein metrics.

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22

Benayadi, Saïd, and Hicham Lebzioui. "Flat left-invariant pseudo-Riemannian metrics on quadratic Lie groups." Journal of Algebra 593 (March 2022): 1–25. http://dx.doi.org/10.1016/j.jalgebra.2021.11.010.

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23

Kankaanrinta, Marja. "Proper smooth G-manifolds have complete G-invariant Riemannian metrics." Topology and its Applications 153, no. 4 (November 2005): 610–19. http://dx.doi.org/10.1016/j.topol.2005.01.034.

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Lebzioui, Hicham. "Flat left-invariant pseudo-Riemannian metrics on unimodular Lie groups." Proceedings of the American Mathematical Society 148, no. 4 (October 28, 2019): 1723–30. http://dx.doi.org/10.1090/proc/14808.

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Duan, Xiaomin, Xueting Ji, Huafei Sun, and Hao Guo. "A Non-Iterative Method for the Difference of Means on the Lie Group of Symmetric Positive-Definite Matrices." Mathematics 10, no. 2 (January 14, 2022): 255. http://dx.doi.org/10.3390/math10020255.

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A non-iterative method for the difference of means is presented to calculate the log-Euclidean distance between a symmetric positive-definite matrix and the mean matrix on the Lie group of symmetric positive-definite matrices. Although affine-invariant Riemannian metrics have a perfect theoretical framework and avoid the drawbacks of the Euclidean inner product, their complex formulas also lead to sophisticated and time-consuming algorithms. To make up for this limitation, log-Euclidean metrics with simpler formulas and faster calculations are employed in this manuscript. Our new approach is to transform a symmetric positive-definite matrix into a symmetric matrix via logarithmic maps, and then to transform the results back to the Lie group through exponential maps. Moreover, the present method does not need to compute the mean matrix and retains the usual Euclidean operations in the domain of matrix logarithms. In addition, for some randomly generated positive-definite matrices, the method is compared using experiments with that induced by the classical affine-invariant Riemannian metric. Finally, our proposed method is applied to denoise the point clouds with high density noise via the K-means clustering algorithm.

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26

CALVARUSO, G., and D. PERRONE. "HOMOGENEOUS AND H-CONTACT UNIT TANGENT SPHERE BUNDLES." Journal of the Australian Mathematical Society 88, no. 3 (May 12, 2010): 323–37. http://dx.doi.org/10.1017/s1446788710000157.

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AbstractWe prove that all g-natural contact metric structures on a two-point homogeneous space are homogeneous contact. The converse is also proved for metrics of Kaluza–Klein type. We also show that if (M,g) is an Einstein manifold and $\tilde G$ is a Riemannian g-natural metric on T1M of Kaluza–Klein type, then $(T_1 M,\tilde \eta ,\tilde G)$ is H-contact if and only if (M,g) is 2-stein, so proving that the main result of Chun et al. [‘H-contact unit tangent sphere bundles of Einstein manifolds’, Q. J. Math., to appear. DOI: 10.1093/qmath/hap025] is invariant under a two-parameter deformation of the standard contact metric structure on T1M. Moreover, we completely characterize Riemannian manifolds admitting two distinct H-contact g-natural contact metric structures, with associated metric of Kaluza–Klein type.

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Besson, Gérard, Gilles Courtois, and Sylvestre Gallot. "Minimal entropy and Mostow's rigidity theorems." Ergodic Theory and Dynamical Systems 16, no. 4 (August 1996): 623–49. http://dx.doi.org/10.1017/s0143385700009019.

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Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y ∈ , we definewhere B(y, R) denotes the ball of radius R around y in . It is a well known fact that this limit exists and does not depend on y ([Man]). The invariant h(g) is called the volume entropy of the metric g but, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sensitive to changes of scale which makes it a bad invariant: however, this is a minor drawback that can be circumvented by looking at the behaviour of the entropy functional on the space of metrics with fixed volume (equal to one for example). Nevertheless, it seems very unlikely that two numbers, the entropy and the volume, might characterize any metric. The very first person to consider such a possibility was Katok ([Kat1]). In this article the entropy is thought of as a dynamical invariant which actually is suggested by its name. More precisely, let us define this dynamical invariant, which is called the topological entropy: let (M, d) be a compact metric space and ψt, a flow on it, we define.

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Liu, Xianfu, and Zuoqin Wang. "On isospectral compactness in conformal class for 4-manifolds." Communications in Contemporary Mathematics 21, no. 05 (July 12, 2019): 1850041. http://dx.doi.org/10.1142/s0219199718500414.

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Let [Formula: see text] be a closed 4-manifold with positive Yamabe invariant and with [Formula: see text]-small Weyl curvature tensor. Let [Formula: see text] be any metric in the conformal class of [Formula: see text] whose scalar curvature is [Formula: see text]-close to a constant. We prove that the set of Riemannian metrics in the conformal class [Formula: see text] that are isospectral to [Formula: see text] is compact in the [Formula: see text] topology.

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Neff, Patrizio, and Robert J. Martin. "Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics." Journal of Geometric Mechanics 8, no. 3 (September 2016): 323–57. http://dx.doi.org/10.3934/jgm.2016010.

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Capogna, Luca, Giovanna Citti, and Maria Manfredini. "Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups." Analysis and Geometry in Metric Spaces 1 (August 6, 2013): 255–75. http://dx.doi.org/10.2478/agms-2013-0006.

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Abstract In this paper we study heat kernels associated with a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics σε which converge in the Gromov- Hausdorff sense to a sub-Riemannian structure on G as ε→ 0. The main new contribution are Gaussian-type bounds on the heat kernel for the σε metrics which are stable as ε→0 and extend the previous time-independent estimates in [16]. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in a step two Carnot group (G; σε ). We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as ε → 0. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow (ε = 0), which in turn yield sub-Riemannian minimal surfaces as t → ∞.

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Balashchenko, Vitaly V., and Anna Sakovich. "Invariantf-structures on the flag manifoldsSO(n)/SO(2)×SO(n−3)." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–15. http://dx.doi.org/10.1155/ijmms/2006/89545.

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We consider manifolds of oriented flagsSO(n)/SO(2)×SO(n−3)(n≥4)as4- and6-symmetric spaces and indicate characteristic conditions for invariant Riemannian metrics under which the canonicalf-structures on these homogeneousΦ-spaces belong to the classesKill f,NKf, andG1fof generalized Hermitian geometry.

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HASHINAGA, Takahiro, Hiroshi TAMARU, and Kazuhiro TERADA. "Milnor-type theorems for left-invariant Riemannian metrics on Lie groups." Journal of the Mathematical Society of Japan 68, no. 2 (April 2016): 669–84. http://dx.doi.org/10.2969/jmsj/06820669.

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Matsumoto, Koji, and Gabriel Teodor Pripoae. "Examples of invariant semi-Riemannian metrics on 4-dimensional lie groups." Rendiconti del Circolo Matematico di Palermo 52, no. 3 (October 2003): 351–66. http://dx.doi.org/10.1007/bf02872760.

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NIKONOROV, YU G. "ON LEFT-INVARIANT EINSTEIN RIEMANNIAN METRICS THAT ARE NOT GEODESIC ORBIT." Transformation Groups 24, no. 2 (January 15, 2018): 511–30. http://dx.doi.org/10.1007/s00031-018-9476-7.

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Bahayou, Amine, and Mohamed Boucetta. "Multiplicative noncommutative deformations of left invariant Riemannian metrics on Heisenberg groups." Comptes Rendus Mathematique 347, no. 13-14 (July 2009): 791–96. http://dx.doi.org/10.1016/j.crma.2009.04.013.

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KOBAYASHI, Osamu. "On a conformally invariant functional of the space of Riemannian metrics." Journal of the Mathematical Society of Japan 37, no. 3 (July 1985): 373–89. http://dx.doi.org/10.2969/jmsj/03730373.

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Nimpa, R. Pefoukeu, M. B. Djiadeu Ngaha, and J. Kamga Wouafo. "Locally symmetric left invariant Riemannian metrics on 3-dimensional Lie groups." Mathematische Nachrichten 290, no. 14-15 (July 31, 2017): 2341–55. http://dx.doi.org/10.1002/mana.201600332.

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Mrowka, Tomasz, Daniel Ruberman, and Nikolai Saveliev. "An index theorem for end-periodic operators." Compositio Mathematica 152, no. 2 (September 7, 2015): 399–444. http://dx.doi.org/10.1112/s0010437x15007502.

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We extend the Atiyah, Patodi, and Singer index theorem for first-order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes’ Fredholm theory for general end-periodic operators. Our index theorem is expressed in terms of a new periodic eta-invariant that equals the Atiyah–Patodi–Singer eta-invariant in the cylindrical setting. We apply this periodic eta-invariant to the study of moduli spaces of Riemannian metrics of positive scalar curvature.

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DUNAJSKI, MACIEJ, and PAUL TOD. "Four–dimensional metrics conformal to Kähler." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 3 (January 5, 2010): 485–503. http://dx.doi.org/10.1017/s030500410999048x.

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AbstractWe derive some necessary conditions on a Riemannian metric (M, g) in four dimensions for it to be locally conformal to Kähler. If the conformal curvature is non anti–self–dual, the self–dual Weyl spinor must be of algebraic type D and satisfy a simple first order conformally invariant condition which is necessary and sufficient for the existence of a Kähler metric in the conformal class. In the anti–self–dual case we establish a one to one correspondence between Kähler metrics in the conformal class and non–zero parallel sections of a certain connection on a natural rank ten vector bundle over M. We use this characterisation to provide examples of ASD metrics which are not conformal to Kähler.We establish a link between the ‘conformal to Kähler condition’ in dimension four and the metrisability of projective structures in dimension two. A projective structure on a surface U is metrisable if and only if the induced (2, 2) conformal structure on M = TU admits a Kähler metric or a para–Kähler metric.

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40

MANGASULI, ANANDATEERTHA. "ON THE EIGENVALUES OF THE LAPLACIAN FOR LEFT-INVARIANT RIEMANNIAN METRICS ON S3." International Journal of Mathematics 18, no. 08 (September 2007): 895–901. http://dx.doi.org/10.1142/s0129167x07004382.

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We show that if the Ricci curvature of a left-invariant metric g on S3 is greater than that of the standard metric g0, then the eigenvalues of Δg are greater than the corresponding eigenvalues of Δgo.

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41

Popiel, Tomasz, and Lyle Noakes. "Elastica in SO(3)." Journal of the Australian Mathematical Society 83, no. 1 (August 2007): 105–24. http://dx.doi.org/10.1017/s1446788700036417.

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AbstractIn a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group SO(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = SO(3), we solve the Euler-Lagrange equation by quadratures.

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42

Kubo, Akira, Kensuke Onda, Yuichiro Taketomi, and Hiroshi Tamaru. "On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups." Hiroshima Mathematical Journal 46, no. 3 (November 2016): 357–74. http://dx.doi.org/10.32917/hmj/1487991627.

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43

ASSELLE, LUCA, and FELIX SCHMÄSCHKE. "On geodesic flows with symmetries and closed magnetic geodesics on orbifolds." Ergodic Theory and Dynamical Systems 40, no. 6 (November 20, 2018): 1480–509. http://dx.doi.org/10.1017/etds.2018.122.

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Let $Q$ be a closed manifold admitting a locally free action of a compact Lie group $G$. In this paper, we study the properties of geodesic flows on $Q$ given by suitable G-invariant Riemannian metrics. In particular, we will be interested in the existence of geodesics that are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.

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44

ARVANITOYEORGOS, ANDREAS, IOANNIS CHRYSIKOS, and YUSUKE SAKANE. "HOMOGENEOUS EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH FIVE ISOTROPY SUMMANDS." International Journal of Mathematics 24, no. 10 (September 2013): 1350077. http://dx.doi.org/10.1142/s0129167x13500778.

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We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad (K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E6/(SU(4) × SU(2) × U(1) × U(1)) and E7/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO (2ℓ + 1)/( U (1) × U (p) × SO (2(ℓ - p - 1) + 1)) and SO (2ℓ)/( U (1) × U (p) × SO (2(ℓ - p - 1))) we prove existence of at least two non-Kähler–Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.

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45

Berestovskii, V. N., and D. E. Vol'per. "A class ofU(n)-invariant Riemannian metrics on manifolds diffeomorphic to odd-dimensional spheres." Siberian Mathematical Journal 34, no. 4 (1993): 612–19. http://dx.doi.org/10.1007/bf00975161.

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46

Belta, C., and V. Kumar. "Euclidean metrics for motion generation on SE(3)." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 216, no. 1 (January 1, 2002): 47–60. http://dx.doi.org/10.1243/0954406021524909.

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Previous approaches to trajectory generation for rigid bodies have been either based on the so-called invariant screw motions or on ad hoc decompositions into rotations and translations. This paper formulates the trajectory generation problem in the framework of Lie groups and Riemannian geometry. The goal is to determine optimal curves joining given points with appropriate boundary conditions on the Euclidean group. Since this results in a two-point boundary value problem that has to be solved iteratively, a computationally efficient, analytical method that generates near-optimal trajectories is derived. The method consists of two steps. The first step involves generating the optimal trajectory in an ambient space, while the second step is used to project this trajectory onto the Euclidean group. The paper describes the method, its applications and its performance in terms of optimality and efficiency.

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STEINBAUER, R., and M. KUNZINGER. "GENERALISED PSEUDO-RIEMANNIAN GEOMETRY FOR GENERAL RELATIVITY." International Journal of Modern Physics A 17, no. 20 (August 10, 2002): 2776. http://dx.doi.org/10.1142/s0217751x0201203x.

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The study of singular spacetimes by distributional methods faces the fundamental obstacle of the inherent nonlinearity of the field equations. Staying strictly within the distributional (in particular: linear) regime, as determined by Geroch and Traschen2 excludes a number of physically interesting examples (e.g., cosmic strings). In recent years, several authors have therefore employed nonlinear theories of generalized functions (Colombeau algebras, in particular) to tackle general relativistic problems1,5,8. Under the influence of these applications in general relativity the nonlinear theory of generalized functions itself has undergone a rapid development lately, resulting in a diffeomorphism invariant global theory of nonlinear generalized functions on manifolds3,4,6. In particular, a generalized pseudo-Riemannian geometry allowing for a rigorous treatment of generalized (distributional) spacetime metrics has been developed7. It is the purpose of this talk to present these new mathematical methods themselves as well as a number of applications in mathematical relativity.

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OVANDO, GABRIELA. "TWO-STEP NILPOTENT LIE ALGEBRAS WITH AD-INVARIANT METRICS AND A SPECIAL KIND OF SKEW-SYMMETRIC MAPS." Journal of Algebra and Its Applications 06, no. 06 (December 2007): 897–917. http://dx.doi.org/10.1142/s0219498807002557.

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We prove that a 2-step nilpotent Lie algebra admitting an ad-invariant metric can be constructed from a vector space 𝔳 endowed with an inner product 〈,〉 and an injective homomorphism ρ : 𝔳 → 𝔰𝔬(𝔳) satisfying ρ(v)v = 0 for all v ∈ 𝔳. The corresponding simply connected pseudo-Riemannian Lie groups are flat and any isometry fixing the identity element does not depend on ρ. The description allows one to construct examples starting with a compact semisimple Lie algebra and it is useful to show some applications.

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SALVAI, MARCOS. "ON THE GEOMETRY OF THE SPACE OF ORIENTED LINES OF THE HYPERBOLIC SPACE." Glasgow Mathematical Journal 49, no. 2 (May 2007): 357–66. http://dx.doi.org/10.1017/s0017089507003710.

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AbstractLet H be the n-dimensional hyperbolic space of constant sectional curvature –1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space $\mathcal{G}_{n}$ of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of $ \mathcal{G}_{n}$ and H. Moreover, we show that $\mathcal{G}_{3}$ is Kähler and find an orthogonal almost complex structure on $\mathcal{G} _{7}$.

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50

Klepikov, P. N. "Left-invariant pseudo-Riemannian metrics on four-dimensional lie groups with nonzero Schouten–Weyl tensor." Russian Mathematics 61, no. 8 (July 19, 2017): 81–85. http://dx.doi.org/10.3103/s1066369x17080102.

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