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Journal articles on the topic 'Generalised flag manifolds'

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Author: Grafiati

Published: 10 March 2023

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1

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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2

Yadav, S., and D. L. Suthar. "On Kenmatsu manifolds Satisfying Certain Conditions." Journal of the Tensor Society 3, no. 00 (June 30, 2009): 19–26. http://dx.doi.org/10.56424/jts.v3i01.9968.

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In this paper, we study 3-dimensional Kenmostu manifolds, weakely Ricci symmetric Kenmostu manifolds and generalized Ricci recurrent Kenmostu manifolds and prove that conformably flat Kenmostu manifold is -Einstein manifolds, deduced that the square length of Ricci tensor. Further proved that if weakly Ricci-symmetric Kenmostu manifolds satisfies Ricci symmetric condition then manifolds Einstein manifold. In last we prove that if generalized Ricci recurrent Kenmostu manifolds satisfies the condition ( )(Y)=0 then .

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3

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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4

De, Uday Chand, Abdallah Abdelhameed Syied, Nasser Bin Turki, and Suliman Alsaeed. "A Study of Generalized Projective P − Curvature Tensor on Warped Product Manifolds." Journal of Mathematics 2021 (December 27, 2021): 1–10. http://dx.doi.org/10.1155/2021/7882356.

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The main aim of this study is to investigate the effects of the P − curvature flatness, P − divergence-free characteristic, and P − symmetry of a warped product manifold on its base and fiber (factor) manifolds. It is proved that the base and the fiber manifolds of the P − curvature flat warped manifold are Einstein manifold. Besides that, the forms of the P − curvature tensor on the base and the fiber manifolds are obtained. The warped product manifold with P − divergence-free characteristic is investigated, and amongst many results, it is proved that the factor manifolds are of constant scalar curvature. Finally, P − symmetric warped product manifold is considered.

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5

Haseeb, Abdul, and Rajendra Prasad. "Certain results on Lorentzian para-Kenmotsu manifolds." Boletim da Sociedade Paranaense de Matemática 39, no. 3 (January 1, 2021): 201–20. http://dx.doi.org/10.5269/bspm.40607.

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The object of the present paper is to study Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection. First we study Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection satisfying the conditions $\bar R\cdot \bar S=0$ and $\bar S\cdot \bar R=0$. After that we study $\phi$-conformally flat, $\phi$-conharmonically flat, $\phi$-concircularly flat, $\phi$-projectively flat and conformally flat Lorentzian para-Kenmotsu manifolds with respect to the quarter-symmetric metric connection and it is shown that in each of these case the manifold is generalized $\eta$-Einstein manifold.

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6

Shenawy, Sameh, and Bülent Ünal. "The W2-curvature tensor on warped product manifolds and applications." International Journal of Geometric Methods in Modern Physics 13, no. 07 (July 25, 2016): 1650099. http://dx.doi.org/10.1142/s0219887816500997.

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The purpose of this paper is to study the [Formula: see text]-curvature tensor on (singly) warped product manifolds as well as on generalized Robertson–Walker and standard static space-times. Some different expressions of the [Formula: see text]-curvature tensor on a warped product manifold in terms of its relation with [Formula: see text]-curvature tensor on the base and fiber manifolds are obtained. Furthermore, we investigate [Formula: see text]-curvature flat warped product manifolds. Many interesting results describing the geometry of the base and fiber manifolds of a [Formula: see text]-curvature flat warped product manifold are derived. Finally, we study the [Formula: see text]-curvature tensor on generalized Robertson–Walker and standard static space-times; we explore the geometry of the fiber of these warped product space-time models that are [Formula: see text]-curvature flat.

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7

ARVANITOYEORGOS, ANDREAS, and IOANNIS CHRYSIKOS. "INVARIANT EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH TWO ISOTROPY SUMMANDS." Journal of the Australian Mathematical Society 90, no. 2 (April 2011): 237–51. http://dx.doi.org/10.1017/s1446788711001303.

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AbstractLet M=G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.

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8

Nagaraja, H. G., and C. R. Premalatha. "Da-Homothetic Deformation of K-Contact Manifolds." ISRN Geometry 2013 (December 16, 2013): 1–7. http://dx.doi.org/10.1155/2013/392608.

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We study Da-homothetic deformations of K-contact manifolds. We prove that Da-homothetically deformed K-contact manifold is a generalized Sasakian space form if it is conharmonically flat. Further, we find expressions for scalar curvature of Da-homothetically deformed K-contact manifolds.

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9

RANDJBAR-DAEMI, S., and J. STRATHDEE. "THE RENORMAUZATION GROUP FOR FLAG MANIFOLDS." International Journal of Modern Physics A 08, no. 20 (August 10, 1993): 3509–28. http://dx.doi.org/10.1142/s0217751x93001417.

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The renormalization group equations for a class of nonrelativistic quantum σ-models targeted on flag manifolds are given. These models emerge in a continuum limit of generalized Heisenberg antiferromagnets. The case of the [Formula: see text] manifold is studied in greater detail. We show that at zero temperature there is a fixed point of the RG transformations in (2 + ε) dimensions where the theory becomes relativistic. We study the linearized RG transformations in the vicinity of this fixed point and show that half of the couplings are irrelevant. We also show that at this fixed point there is an enlargement of the global isometries of the target manifold. We construct a discrete non-Abelian enlargement of this kind.

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10

Zhuang, Xiaobo. "Vanishing theorems of generalized Witten genus for generalized complete intersections in flag manifolds." International Journal of Mathematics 27, no. 09 (August 2016): 1650076. http://dx.doi.org/10.1142/s0129167x16500762.

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We propose a potential function [Formula: see text] for the cohomology ring of partial flag manifolds. We prove a formula expressing integrals over partial flag manifolds by residues, which generalizes [E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in Geometry, Topology, Physics (International Press, 1995), pp. 357–422]. Using this formula, we prove a Landweber–Stong type vanishing theorem for generalized [Formula: see text] complete intersections in flag manifolds, which serves as evidence for the [Formula: see text] version of Stolz conjecture [Q. Chen, F. Han and W. Zhang, Generalized Witten genus and vanishing theorems, J. Differential Geom. 88(1) (2011) 1–39].

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11

BONNEAU, GUY, and FRANÇOIS DELDUC. "CONSTRUCTION AND PROPERTIES OF QUASI RICCI FLAT SPACES." International Journal of Modern Physics A 01, no. 04 (December 1986): 997–1007. http://dx.doi.org/10.1142/s0217751x86000381.

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We look for the necessary and sufficient conditions for a generalized torsion-free nonlinear σ-model to be one-loop finite. The corresponding metrics are not only Ricci flat ones, but also a larger class we call “quasi Ricci flat” spaces. We give expressions for the corresponding Lagrangian densities in the real and Kähler cases. In the latter, the manifold is shown to be proper, complete and nonhomogeneous. Unfortunately, in the compact case, relevant for string theory, these quasi Ricci flat manifolds become Ricci flat ones.

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12

Cahen, Benjamin. "Berezin quantization on generalized flag manifolds." MATHEMATICA SCANDINAVICA 105, no. 1 (September 1, 2009): 66. http://dx.doi.org/10.7146/math.scand.a-15106.

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Let $M=G/H$ be a generalized flag manifold where $G$ is a compact, connected, simply-connected Lie group with Lie algebra $\mathfrak{g}$ and $H$ is the centralizer of a torus. Let $\pi$ be a unitary irreducible representation of $G$ which is holomorphically induced from a character of $H$. Using a complex parametrization of a dense open subset of $M$, we realize $\pi$ on a Hilbert space of holomorphic functions. We give explicit expressions for the differential $d\pi$ of $\pi$ and for the Berezin symbols of $\pi (g)$ ($g\in G$) and $d\pi (X)$ ($X\in \mathfrak{g}$). In particular, we recover some results of S. Berceanu and we partially generalize a result of K. H. Neeb.

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13

Puerta, X. "Versal deformations in generalized flag manifolds." Linear Algebra and its Applications 401 (May 2005): 445–51. http://dx.doi.org/10.1016/j.laa.2004.11.029.

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14

Blazic, Novica. "The volumes of small geodesic balls and generalized Chern numbers of Kaehler manifolds." Nagoya Mathematical Journal 116 (December 1989): 181–89. http://dx.doi.org/10.1017/s0027763000001768.

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In this paper we study a connection between global and local properties of Kaehler manifolds, more specifically we study a connection between the volumes of small geodesic balls of a manifold M and some generalized Chern numbers. We use the standard power series expansion for Vm(r).In Theorem 3.1 we give characterizations of a flat compact Kaehler manifold in terms of the volumes of small geodesic balls and generalized Chern numbers ωn-1c1(M) and ωn-2c12(M). In Theorem 4.1 similar questions for complex space forms are considered. So we prove one particular case of the Conjecture (IV) stated by Gray and Vanhecke [6].

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15

Cortés, Vicente, and Liana David. "Twist, elementary deformation and K/K correspondence in generalized geometry." International Journal of Mathematics 31, no. 10 (September 2020): 2050078. http://dx.doi.org/10.1142/s0129167x20500780.

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We define the conformal change and elementary deformation in generalized complex geometry. We apply Swann’s twist construction to generalized (almost) complex and Hermitian structures obtained by these operations and establish conditions for the Courant integrability of the resulting twisted structures. We associate to any appropriate generalized Kähler manifold [Formula: see text] with a Hamiltonian Killing vector field a new generalized Kähler manifold, depending on the choice of a pair of non-vanishing functions and compatible twist data. We study this construction when [Formula: see text] is toric, with emphasis on the four-dimensional case, and we apply it to deformations of the standard flat Kähler metric on [Formula: see text], the Fubini–Study metric on [Formula: see text] and the admissible Kähler metrics on Hirzebruch surfaces. As a further application, we recover the K/K (Kähler/Kähler) correspondence, by specializing to ordinary Kähler manifolds.

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16

Bukusheva, A. V. "Non-holonomic Kenmotsu manifolds equipped with generalized Tanaka — Webster connection." Differential Geometry of Manifolds of Figures, no. 52 (2021): 42–51. http://dx.doi.org/10.5922/0321-4796-2020-52-5.

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А non-holonomic Kenmotsu manifold equipped with a connection analogous to the generalized Tanaka — Webster connection, is considered. The studied connection is obtained from the generalized Tanaka — Webster connection by replacing the first structural endomorphism by the second structural endomorphism. The obtained connection is also called in the work the generalized Tanaka — Webster connection. Unlike a Kenmotsu manifold, the structure form of a non-holonomic Kenmotsu manifold is not closed. The consequence of this single difference is a significant discrepancy in the properties of such manifolds. For example, it is proved in the paper that the alternation of the Ricci-Schouten tensor of a non-holonomic Kenmotsu manifold, which is a transverse analogue of the Ricci tensor, is proportional to the external differential of the structural form. At the same time, in the classical case of a Kenmotsu manifold, the Ricci — Schouten tensor is a symmetric tensor. It is proved that a Tanaka — Webster connection is a metric connection. It is also proved that from the fact that the alternation of the Ricci-Schouten tensor is proportional to the external differential of the structural form, the following statement holds: if a non-holonomic Kenmotsu manifold is an Einstein manifold with respect to the generalized Tanaka — Webster connection, then it is Ricci-flat with respect to the same connection.

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17

Zixin, Hou. "On harmonic maps between generalized flag manifolds." Acta Mathematica Sinica 8, no. 1 (March 1992): 1–16. http://dx.doi.org/10.1007/bf02595015.

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18

Varea, Carlos A. B., and Luiz A. B. San Martin. "Invariant generalized complex structures on flag manifolds." Journal of Geometry and Physics 150 (April 2020): 103610. http://dx.doi.org/10.1016/j.geomphys.2020.103610.

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19

Guest, M. A. "Geometry of maps between generalized flag manifolds." Journal of Differential Geometry 25, no. 2 (1987): 223–47. http://dx.doi.org/10.4310/jdg/1214440851.

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20

Snow, Dennis M., and Kirk Weller. "A vanishing theorem for generalized flag manifolds." Archiv der Mathematik 64, no. 5 (May 1995): 444–51. http://dx.doi.org/10.1007/bf01197223.

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21

Deshmukh, Sharief, and Ibrahim Al-Dayel. "Concircularity on GRW-space-times and conformally flat spaces." International Journal of Geometric Methods in Modern Physics 18, no. 08 (May 8, 2021): 2150132. http://dx.doi.org/10.1142/s0219887821501322.

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There are two smooth functions [Formula: see text] and [Formula: see text] associated to a nontrivial concircular vector field [Formula: see text] on a connected Riemannian manifold [Formula: see text], called potential function and connecting function. In this paper, we show that presence of a timelike nontrivial concircular vector field influences the geometry of generalized Robertson–Walker space-times. We use a timelike concircular vector field [Formula: see text] on an [Formula: see text] -dimensional connected conformally flat Lorentzian manifold, [Formula: see text], to find a characterization of generalized Robertson–Walker space-time with fibers Einstein manifolds. It is interesting to note that for [Formula: see text] the concircular vector field annihilates energy-momentum tensor and also that in this case the potential function [Formula: see text] is harmonic. In the second part of this paper, we show that presence of a nontrivial concircular vector field [Formula: see text] with connecting function [Formula: see text] on a complete and connected [Formula: see text] -dimensional conformally flat Riemannian manifold [Formula: see text], [Formula: see text], with Ricci curvature [Formula: see text] non-negative, satisfying [Formula: see text], is necessary and sufficient for [Formula: see text] to be isometric to either a sphere [Formula: see text] or to the Euclidean space [Formula: see text], where [Formula: see text] is the scalar curvature.

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22

Prakasha, Doddabhadrappla G., Luis M. Fernández, and Kakasab Mirji. "The ℳ-projective curvature tensor field on generalized (κ,μ)-paracontact metric manifolds." Georgian Mathematical Journal 27, no. 1 (March 1, 2020): 141–47. http://dx.doi.org/10.1515/gmj-2017-0054.

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AbstractWe consider generalized {(\kappa,\mu)}-paracontact metric manifolds satisfying certain flatness conditions on the {\mathcal{M}}-projective curvature tensor. Specifically, we study ξ-{\mathcal{M}}-projectively flat and {\mathcal{M}}-projectively flat generalized {(\kappa,\mu)}-paracontact metric manifolds and, further, ϕ-{\mathcal{M}}-projectively symmetric generalized {(\kappa\neq-1,\mu)}-paracontact metric manifolds. We prove that they are characterized by certain structures whose properties are discussed in some detail.

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23

Arvanitoyeorgos, Andreas. "New invariant Einstein metrics on generalized flag manifolds." Transactions of the American Mathematical Society 337, no. 2 (February 1, 1993): 981–95. http://dx.doi.org/10.1090/s0002-9947-1993-1097162-3.

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24

Stokman, Jasper V. "The quantum orbit method for generalized flag manifolds." Mathematical Research Letters 10, no. 4 (2003): 469–81. http://dx.doi.org/10.4310/mrl.2003.v10.n4.a6.

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25

Varea, Carlos A. B. "Invariant generalized complex structures on partial flag manifolds." Indagationes Mathematicae 31, no. 4 (July 2020): 536–55. http://dx.doi.org/10.1016/j.indag.2020.04.003.

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26

Li, Ping, and Yang Su. "The signature of generalized flag manifolds and applications." Acta Mathematica Sinica, English Series 26, no. 8 (July 15, 2010): 1457–62. http://dx.doi.org/10.1007/s10114-010-9014-6.

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27

Yamada, Takumi. "Invariant pseudo-Kähler metrics on generalized flag manifolds." Differential Geometry and its Applications 36 (October 2014): 44–55. http://dx.doi.org/10.1016/j.difgeo.2014.07.003.

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28

Moroianu, Andrei, and Uwe Semmelmann. "Generalized Killing spinors on Einstein manifolds." International Journal of Mathematics 25, no. 04 (April 2014): 1450033. http://dx.doi.org/10.1142/s0129167x14500335.

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We study generalized Killing spinors on compact Einstein manifolds with positive scalar curvature. This problem is related to the existence of compact Einstein hypersurfaces in manifolds with parallel spinors, or equivalently, in Riemannian products of flat spaces, Calabi–Yau, hyperkähler, G2 and Spin(7) manifolds.

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29

Segev, Reuven. "Continuum mechanics, stresses, currents and electrodynamics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374, no. 2066 (April 28, 2016): 20150174. http://dx.doi.org/10.1098/rsta.2015.0174.

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The Eulerian approach to continuum mechanics does not make use of a body manifold. Rather, all fields considered are defined on the space, or the space–time, manifolds. Sections of some vector bundle represent generalized velocities which need not be associated with the motion of material points. Using the theories of de Rham currents and generalized sections of vector bundles, we formulate a weak theory of forces and stresses represented by vector-valued currents. Considering generalized velocities represented by differential forms and interpreting such a form as a generalized potential field, we present a weak formulation of pre-metric, p -form electrodynamics as a natural example of the foregoing theory. Finally, it is shown that the assumptions leading to p -form electrodynamics may be replaced by the condition that the force functional is continuous with respect to the flat topology of forms.

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30

Kuroki, Shintarô, Eunjeong Lee, Jongbaek Song, and Dong Youp Suh. "Flag Bott manifolds and the toric closure of a generic orbit associated to a generalized Bott manifold." Pacific Journal of Mathematics 308, no. 2 (December 9, 2020): 347–92. http://dx.doi.org/10.2140/pjm.2020.308.347.

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31

Nishinou, Takeo. "Toric Degenerations, Tropical Curve, and Gromov–Witten Invariants of Fano Manifolds." Canadian Journal of Mathematics 67, no. 3 (June 1, 2015): 667–95. http://dx.doi.org/10.4153/cjm-2014-006-3.

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AbstractIn this paper, we give a tropical method for computing Gromov–Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds that admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A and some moduli space of rank two bundles on a genus two curve.

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32

Grama, Lino, and Caio J. C. Negreiros. "Equigeodesics on Generalized Flag Manifolds with Two Isotropy Summands." Results in Mathematics 60, no. 1-4 (June 2, 2011): 405–21. http://dx.doi.org/10.1007/s00025-011-0149-2.

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33

Mare, Augustin-Liviu. "Quantum cohomology of the infinite-dimensional generalized flag manifolds." Advances in Mathematics 185, no. 2 (July 2004): 347–69. http://dx.doi.org/10.1016/j.aim.2003.07.005.

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34

Gardner, Robert B., Marcus Kriele, and Udo Simon. "Generalized spherical functions on projectively flat manifolds." Results in Mathematics 27, no. 1-2 (March 1995): 41–50. http://dx.doi.org/10.1007/bf03322268.

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35

David, Liana. "The Bochner-flat cone of a CR manifold." Compositio Mathematica 144, no. 3 (May 2008): 747–73. http://dx.doi.org/10.1112/s0010437x07003363.

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AbstractWe construct a Kähler structure (which we call a generalised Kähler cone) on an open subset of the cone of a strongly pseudo-convex CR manifold endowed with a one-parameter family of compatible Sasaki structures. We determine those generalised Kähler cones which are Bochner-flat and we study their local geometry. We prove that any Bochner-flat Kähler manifold of complex dimension bigger than two is locally isomorphic to a generalised Kähler cone.

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36

Shaikh, A. A., and Sanjib Kumar Jana. "On pseudo generalized quasi-Einstein manifolds." Tamkang Journal of Mathematics 39, no. 1 (March 31, 2008): 9–24. http://dx.doi.org/10.5556/j.tkjm.39.2008.41.

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The object of the present paper is to introduce a type of non-flat Riemannian man- ifold called pseudo generalized quasi-Einstein manifold and studied some properties of such a manifold with several non-trivial examples.

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37

Pal, Buddhadev, Santu Dey, and Sampa Pahan. "On a non flat Riemannian warped product manifold with respect to quarter-symmetric connection." Acta Universitatis Sapientiae, Mathematica 11, no. 2 (December 1, 2019): 332–49. http://dx.doi.org/10.2478/ausm-2019-0024.

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Abstract In this paper, we study generalized quasi-Einstein warped products with respect to quarter symmetric connection for dimension n ≥ 3 and Ricci-symmetric generalized quasi-Einstein manifold with quarter symmetric connection. We also investigate that in what conditions the generalized quasi-Einstein manifold to be nearly Einstein manifold with respect to quarter symmetric connection. Example of warped product on generalized quasi-Einstein manifold with respect to quarter symmetric connection are also discussed.

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38

., Venkatesha, and S. V. Vishnuvardhana. "τ-Curvature On Kenmotsu Manifold." Journal of the Tensor Society 8, no. 01 (June 30, 2007): 103–11. http://dx.doi.org/10.56424/jts.v8i01.10554.

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In the present paper we have obtained the necessary and sufficient condition for a extended generalized τ -ϕ-recurrent Kenmotsu manifold to be a generalized ricci-recurrent manifold. Furthermore, we have studied τ -ϕ-symmetric Kenmotsu manifold, τ -ξ-flat Kenmotsu manifold and a Kenmotsu manifold satisfying τ (X, Y ) · R = 0.

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39

Calixto, Manuel. "Generalized higher-spin algebras and symbolic calculus on flag manifolds." Journal of Geometry and Physics 56, no. 2 (February 2006): 143–74. http://dx.doi.org/10.1016/j.geomphys.2005.01.003.

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40

Schwarz, Benjamin. "Hilbert series of nearly holomorphic sections on generalized flag manifolds." Advances in Mathematics 290 (February 2016): 293–313. http://dx.doi.org/10.1016/j.aim.2015.12.003.

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41

Dragomir, Sorin, and Renata Grimaldi. "Generalized Hopf manifolds with flat local Kaelher metrics." Annales de la faculté des sciences de Toulouse Mathématiques 10, no. 3 (1989): 361–68. http://dx.doi.org/10.5802/afst.682.

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42

Mikeš, Josef, Vladimir Rovenski, Sergey Stepanov, and Irina Tsyganok. "Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds." Mathematics 9, no. 9 (April 22, 2021): 927. http://dx.doi.org/10.3390/math9090927.

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In this article, we discuss the global aspects of the geometry of locally conformally flat (complete and compact) Riemannian manifolds. In particular, the article reviews and improves some results (e.g., the conditions of compactness and degeneration into spherical or flat space forms) on the geometry “in the large" of locally conformally flat Riemannian manifolds. The results presented here were obtained using the generalized and classical Bochner technique, as well as the Ricci flow.

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43

Arvanitoyeorgos, Andreas, Yusuke Sakane, and Marina Statha. "New homogeneous Einstein metrics on quaternionic Stiefel manifolds." Advances in Geometry 18, no. 4 (October 25, 2018): 509–24. http://dx.doi.org/10.1515/advgeom-2018-0014.

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Abstract We consider invariant Einstein metrics on the quaternionic Stiefel manifold Vpℍn of all orthonormal p-frames in ℍn. This manifold is diffeomorphic to the homogeneous space Sp(n)/Sp(n − p) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on Vpℍn ≅ Sp(n)/Sp(n − p), where n = k1 + k2 + k3 and p = n − k3. We view Vpℍn as a total space over the generalized Wallach space Sp(n)/(Sp(k1)×Sp(k2)×Sp(k3)) and over the generalized flag manifold Sp(n)/(U(p)×Sp(n − p)).

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44

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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45

Yang, Kichoon. "Plücker formulae for the orthogonal group." Bulletin of the Australian Mathematical Society 40, no. 3 (December 1989): 447–56. http://dx.doi.org/10.1017/s0004972700017512.

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Plücker formulae for horizontal curves in SO(m)-flag manifolds are derived. These formulae are seen to generalise the usual Plücker formulae for projective space curves. They also have applications in the theory of minimal surfaces in Euclidean sphere and the complex hyperquadric.

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46

Gasparim, Elizabeth, Fabricio Valencia, and Carlos Varea. "Invariant generalized complex geometry on maximal flag manifolds and their moduli." Journal of Geometry and Physics 163 (May 2021): 104108. http://dx.doi.org/10.1016/j.geomphys.2021.104108.

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47

Wang, Yu, and Guosong Zhao. "Equigeodesics on Generalized Flag Manifolds with b 2 (G / K) = 1." Results in Mathematics 64, no. 1-2 (December 4, 2012): 77–90. http://dx.doi.org/10.1007/s00025-012-0298-y.

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48

Matassa, Marco. "An analogue of Weyl’s law for quantized irreducible generalized flag manifolds." Journal of Mathematical Physics 56, no. 9 (September 2015): 091704. http://dx.doi.org/10.1063/1.4931606.

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49

Marastoni, Corrado, and Toshiyuki Tanisaki. "Radon transforms for quasi-equivariant D-modules on generalized flag manifolds." Differential Geometry and its Applications 18, no. 2 (March 2003): 147–76. http://dx.doi.org/10.1016/s0926-2245(02)00145-6.

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50

Prado, Rafaela F. do, and Lino Grama. "Variational aspects of homogeneous geodesics on generalized flag manifolds and applications." Annals of Global Analysis and Geometry 55, no. 3 (November 3, 2018): 451–77. http://dx.doi.org/10.1007/s10455-018-9635-z.

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