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Journal articles on the topic 'Geometry of null manifolds'

Author: Grafiati

Published: 23 November 2024

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1

Atindogbé, C., M. Gutiérrez, and R. Hounnonkpe. "Compact null hypersurfaces in Lorentzian manifolds." Advances in Geometry 21, no. 2 (April 1, 2021): 251–63. http://dx.doi.org/10.1515/advgeom-2021-0001.

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Abstract We show how the topological and geometric properties of the family of null hypersurfaces in a Lorentzian manifold are related with the properties of the ambient manifold itself. In particular, we focus in how the presence of global symmetries and curvature conditions restrict the existence of compact null hypersurfaces. We use these results to show the influence on the existence of compact totally umbilic null hypersurfaceswhich are not totally geodesic. Finally we describe the restrictions that they impose in causality theory.

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2

Massamba, Fortuné. "Symmetries of Null Geometry in Indefinite Kenmotsu Manifolds." Mediterranean Journal of Mathematics 10, no. 2 (June 24, 2012): 1079–99. http://dx.doi.org/10.1007/s00009-012-0205-5.

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3

Hoffman, Neil R., and Nathan S. Sunukjian. "Null-homologous exotic surfaces in 4–manifolds." Algebraic & Geometric Topology 20, no. 5 (November 4, 2020): 2677–85. http://dx.doi.org/10.2140/agt.2020.20.2677.

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4

Cuadros Valle, Jaime. "Null Sasaki $$\eta $$ -Einstein structures in 5-manifolds." Geometriae Dedicata 169, no. 1 (April 24, 2013): 343–59. http://dx.doi.org/10.1007/s10711-013-9859-9.

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5

Akamine, Shintaro, Atsufumi Honda, Masaaki Umehara, and Kotaro Yamada. "Null hypersurfaces in Lorentzian manifolds with the null energy condition." Journal of Geometry and Physics 155 (September 2020): 103751. http://dx.doi.org/10.1016/j.geomphys.2020.103751.

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6

Rovenski, Vladimir, Sergey Stepanov, and Josef Mikeš. "A Note on the Geometry of Certain Classes of Lichnerowicz Laplacians and Their Applications." Mathematics 11, no. 21 (October 26, 2023): 4434. http://dx.doi.org/10.3390/math11214434.

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In the present paper, we prove vanishing theorems for the null space of the Lichnerowicz Laplacian acting on symmetric two tensors on complete and closed Riemannian manifolds and further estimate its lowest eigenvalue on closed Riemannian manifolds. In addition, we give an application of the obtained results to the theory of infinitesimal Einstein deformations.

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7

Massamba, Fortuné, and Samuel Ssekajja. "Some Remarks on Quasi-Generalized CR-Null Geometry in Indefinite Nearly Cosymplectic Manifolds." International Journal of Mathematics and Mathematical Sciences 2016 (2016): 1–10. http://dx.doi.org/10.1155/2016/9613182.

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Attention is drawn to some distributions on ascreen Quasi-Generalized Cauchy-Riemannian (QGCR) null submanifolds in an indefinite nearly cosymplectic manifold. We characterize totally umbilical and irrotational ascreen QGCR-null submanifolds. We finally discuss the geometric effects of geodesity conditions on such submanifolds.

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8

Massamba, Fortuné, and Samuel Ssekajja. "A geometric flow on null hypersurfaces of Lorentzian manifolds." Topological Algebra and its Applications 10, no. 1 (January 1, 2022): 185–95. http://dx.doi.org/10.1515/taa-2022-0126.

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Abstract We introduce a geometric flow on a screen integrable null hypersurface in terms of its local second fundamental form. We use it to give an alternative proof to the vorticity free Raychaudhuri’s equation for null hypersurface, as well as establishing conditions for the existence of constant mean curvature (CMC) null hypersurfaces, and leaves of constant scalar curvatures.

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9

Duggal, K. L. "A Review on Unique Existence Theorems in Lightlike Geometry." Geometry 2014 (July 7, 2014): 1–17. http://dx.doi.org/10.1155/2014/835394.

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This is a review paper of up-to-date research done on the existence of unique null curves, screen distributions, Levi-Civita connection, symmetric Ricci tensor, and scalar curvature for a large variety of lightlike submanifolds of semi-Riemannian (in particular, Lorentzian) manifolds, supported by examples and an extensive bibliography. We also propose some open problems.

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10

Kim, Jin Hong. "On null cobordism classes of quasitoric manifolds and their small covers." Topology and its Applications 285 (November 2020): 107412. http://dx.doi.org/10.1016/j.topol.2020.107412.

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11

Ssekajja, Samuel. "Remarks on Screen Integrable Null Hypersurfaces in Lorentzian Manifolds." Zurnal matematiceskoj fiziki, analiza, geometrii 16, no. 4 (September 25, 2020): 460–72. http://dx.doi.org/10.15407/mag16.04.460.

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12

Safi, Alamgir, Muhammad Asghar Khan, Muhammad Adnan Aziz, Mohammed H. Alsharif, Tanweer Ahmad Cheema, Insaf Ullah, Abu Jahid, Abdulaziz H. Alghtani, and Ayman A. Aly. "Application of Differential Geometry to the Array Manifolds of Linear Arrays in Antenna Array Processing." Electronics 10, no. 23 (November 28, 2021): 2964. http://dx.doi.org/10.3390/electronics10232964.

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This article deals with the application of differential geometry to the array manifolds of non-uniform linear antenna array (NULA) when estimating the direction of arrival (DOA) of multiple sources present in an environment using far field approximation. In order to resolve this issue, we utilized a doublet linear antenna array (DLA) comprising two individual NULAs, along with a proposed algorithm that chooses correct directions of the impinging sources with the help of the prior knowledge of the ambiguous directions calculated with the application of differential geometry to the manifold curves of each NULA. The algorithm checks the correlation of the estimated direction of arrival (DOAs) by both the individual NULA with its corresponding ambiguous set of directions and chooses the output of the NULA, which has a minimum correlation between their estimated DOAs and corresponding ambiguous DOAs. DLA is designed such that the intersection of all the ambiguous set of DOAs among the individual NULAs are null sets. DOA of sources, which imping signals from different directions on the DLA, are estimated using three direction finding (DF) techniques, such as, genetic algorithm (GA), pattern search (PS), and a hybrid technique that utilizes both GA and PS at the same time. As compared to the existing techniques of ambiguity resolution, the proposed algorithm improves the estimation accuracy. Simulation results for all the three DF techniques utilizing the DLA along with the proposed algorithm are presented using MATLAB. As compared to the genetic algorithm and pattern search, the intelligent hybrid technique, such that, GA–PS, had better estimation accuracy in choosing corrected DOAs, despite the fact that the impinging DOAs were from ambiguous directions.

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13

Popescu, Paul, Vladimir Rovenski, and Sergey Stepanov. "The Weitzenböck Type Curvature Operator for Singular Distributions." Mathematics 8, no. 3 (March 6, 2020): 365. http://dx.doi.org/10.3390/math8030365.

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We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their L 2 adjoint operators on tensors. Then, we introduce the Weitzenböck type curvature operator on tensors, prove the Weitzenböck type decomposition formula, and derive the Bochner–Weitzenböck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones.

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14

Cahn, Patricia, and Alexandra Kjuchukova. "Linking Numbers in Three-Manifolds." Discrete & Computational Geometry 66, no. 2 (July 6, 2021): 435–63. http://dx.doi.org/10.1007/s00454-021-00287-3.

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AbstractLet M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of $$S^3$$ S 3 branched along a knot $$\alpha \subset S^3$$ α ⊂ S 3 . Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot $$\alpha $$ α can be derived from dihedral covers of $$\alpha $$ α . The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications.

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15

Boja, Nicolae. "On the trace-manifold generated by the deformations of a body-manifold." Theoretical and Applied Mechanics, no. 30 (2003): 11–28. http://dx.doi.org/10.2298/tam0301011b.

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In this paper, concerned to the study of continuous deformations of material media using some tools of modem differential geometry, a moving frame of Frenet type along the orbits of an one-parameter group acting on a so-called "trace-manifold", M, associated to the deformations, is constructed. The manifold M is defined as an infinite union of non-disjoint compact manifolds, generated by the consecutive positions in the Euclidean affine 3-space of a body-manifold under deformations in a closed time interval. We put in evidence a skew-symmetric band tensor of second order, ?, which describes the deformation in a small neighborhood of any point along the orbits. The non-null components ?i,i+i, (i =1,2), of ? are assimilated as like curvatures at each point of an orbit in the planes generated by the pairs of vectors (?i,?i+i) of a moving frame in M associated to the orbit in a similar way as the Frenet's frame is. Also a formula for the energy of the orbits is given and its relationship with some stiffness matrices is established.

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16

Lane, Jeremy. "The Geometric Structure of Symplectic Contraction." International Mathematics Research Notices 2020, no. 12 (June 8, 2018): 3521–39. http://dx.doi.org/10.1093/imrn/rny122.

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Abstract We show that the symplectic contraction map of Hilgert–Manon–Martens [9], a symplectic version of Popov’s horospherical contraction, is simply the quotient of a Hamiltonian manifold $M$ by a “stratified null foliation” that is determined by the group action and moment map. We also show that the quotient differential structure on the symplectic contraction of $M$ supports a Poisson bracket. We end by proving a very general description of the topology of fibers of Gelfand–Zeitlin (also spelled Gelfand–Tsetlin or Gelfand–Cetlin) systems on multiplicity-free Hamiltonian $U(n)$ and $SO(n)$ manifolds.

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17

Rovenski, Vladimir, and Dhriti Sundar Patra. "Characteristics of Sasakian Manifolds Admitting Almost ∗-Ricci Solitons." Fractal and Fractional 7, no. 2 (February 4, 2023): 156. http://dx.doi.org/10.3390/fractalfract7020156.

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This article presents some results of a geometric classification of Sasakian manifolds (SM) that admit an almost ∗-Ricci soliton (RS) structure (g,ω,X). First, we show that a complete SM equipped with an almost ∗-RS with ω≠ const is a unit sphere. Then we prove that if an SM has an almost ∗-RS structure, whose potential vector is a Jacobi vector field on the integral curves of the characteristic vector field, then the manifold is a null or positive SM. Finally, we characterize those SM represented as almost ∗-RS, which are ∗-RS, ∗-Einstein or ∗-Ricci flat.

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18

Chaturvedi, S., E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, and R. Simon. "Null phase curves and manifolds in geometric phase theory." Journal of Mathematical Physics 54, no. 6 (June 2013): 062106. http://dx.doi.org/10.1063/1.4811346.

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19

Dey, Tamal K. "Efficient Algorithms to Detect Null-Homologous Cycles on 2-Manifolds." International Journal of Computational Geometry & Applications 07, no. 03 (June 1997): 167–74. http://dx.doi.org/10.1142/s0218195997000119.

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Given a cycle of length k on a triangulated 2-manifold, we determine if it is null-homologous (bounds a surface) in O(n+k) optimal time and space where n is the size of the triangulation. Further, with a preprocessing step of O(n) time we answer the same query for any cycle of length k in O(g+k) time, g the genus of the 2-manifold. This is optimal for k < g.

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20

Abbassi, Mohamed Tahar Kadaoui, Khadija Boulagouaz, and Giovanni Calvaruso. "On the Geometry of the Null Tangent Bundle of a Pseudo-Riemannian Manifold." Axioms 12, no. 10 (September 22, 2023): 903. http://dx.doi.org/10.3390/axioms12100903.

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When we consider a non-definite pseudo-Riemannian manifold, we obtain lightlike tangent vectors that constitute the null tangent bundle, whose fibers are lightlike cones in the corresponding tangent spaces. In this paper, we define and study a class of “g-natural” metrics on the tangent bundle of a pseudo-Riemannian manifold and we investigate the geometry of the null tangent bundle as a lightlike hypersurface equipped with an induced g-natural metric.

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21

Şahin, Bayram. "Warped Product Lightlike Submanifolds." Sarajevo Journal of Mathematics 1, no. 2 (June 12, 2024): 251–60. http://dx.doi.org/10.5644/sjm.01.2.10.

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e study a new class of lightlike submanifolds $M$, called warped product lightlike submanifolds, of a semi-Riemann manifold. We show that the null geometry of $M$ reduces to the corresponding non-degenerate geometry of its semi-Riemann submanifold. 2000 Mathematics Subject Classification. 53C15, 53C40, 53C50

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22

Machida, Yoshinori, and Hajime Sato. "Twistor theory of manifolds with Grassmannian structures." Nagoya Mathematical Journal 160 (2000): 17–102. http://dx.doi.org/10.1017/s0027763000007698.

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AbstractAs a generalization of the conformal structure of type (2, 2), we study Grassmannian structures of type (n, m) forn, m≥ 2. We develop their twistor theory by considering the complete integrability of the associated null distributions. The integrability corresponds to global solutions of the geometric structures.A Grassmannian structure of type (n, m) on a manifoldMis, by definition, an isomorphism from the tangent bundleTMofMto the tensor productV ⊗ Wof two vector bundlesVandWwith ranknandmoverMrespectively. Because of the tensor product structure, we have two null plane bundles with fibresPm-1(ℝ) andPn-1(ℝ) overM. The tautological distribution is defined on each two bundles by a connection. We relate the integrability condition to the half flatness of the Grassmannian structures. Tanaka’s normal Cartan connections are fully used and the Spencer cohomology groups of graded Lie algebras play a fundamental role.Besides the integrability conditions corrsponding to the twistor theory, the lifting theorems and the reduction theorems are derived. We also study twistor diagrams under Weyl connections.

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23

BURDET, GUY, TAXIARCHIS PAPACOSTAS, and MARTINE PERRIN. "LORENTZIAN MANIFOLDS ADMITTING ISOTROPIC HYPERSURFACES SOLUTIONS OF EINSTEIN’S FIELD EQUATIONS." International Journal of Modern Physics D 03, no. 01 (March 1994): 163–66. http://dx.doi.org/10.1142/s0218271894000198.

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All spaces solutions of Einstein’s field equations, admitting an isotropic (null) hypersurface (hereafter referred under the acronymus “ishyps”) are determined in a geometric way. We consider in more details two sub-cases, the generalized Robinson-Bertotti, and the pp-waves spaces.

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24

Duggal, K. L., and B. Sahin. "Screen conformal half-lightlike submanifolds." International Journal of Mathematics and Mathematical Sciences 2004, no. 68 (2004): 3737–53. http://dx.doi.org/10.1155/s0161171204403342.

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We study some properties of a half-lightlike submanifoldM, of a semi-Riemannian manifold, whose shape operator is conformal to the shape operator of its screen distribution. We show that any screen distributionS(TM)ofMis integrable and the geometry ofMhas a close relation with the nondegenerate geometry of a leaf ofS(TM). We prove some results on symmetric induced Ricci tensor and null sectional curvature of this class.

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25

Bergman, David R. "Differential Geometry and Acoustics: A Survey." International Journal of Acoustics and Vibration 26, no. 2 (June 30, 2021): 95–102. http://dx.doi.org/10.20855/ijav.2021.26.21686.

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A connection between acoustic rays in a moving inhomogeneous fluid medium and the null geodesic of a pseudo-Riemannian manifold provides a mechanism to derive several well-known results commonly used in acoustic ray theory. Among these include ray integrals for depth dependent sound speed and current profiles commonly used in ocean and aero acoustic modelling. In this new paradigm these are derived by application of a symmetry of the effective metric tensor known as isometry. In addition to deriving well-known results, the application of the full machinery of differential geometry offers a unified approach to modelling acoustic fields in three dimensional random environments with time dependence by, (1) using conformal symmetry to simplify the geodesic equation, and (2) application of geodesic deviation as a generalization of geometric spread.

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26

Gutiérrez, Manuel, and Benjamín Olea. "Conditions on a null hypersurface of a Lorentzian manifold to be a null cone." Journal of Geometry and Physics 145 (November 2019): 103469. http://dx.doi.org/10.1016/j.geomphys.2019.06.020.

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27

Barbaresco, Frédéric. "Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation." Entropy 22, no. 6 (June 9, 2020): 642. http://dx.doi.org/10.3390/e22060642.

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In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo’s group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.

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28

Tang, Mengjiao, Yao Rong, and Jie Zhou. "An Information Geometric Viewpoint on the Detection of Range Distributed Targets." Mathematical Problems in Engineering 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/930793.

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The paper adopts the information geometry, to put forward a new viewpoint on the detection of range distributed targets embedded in Gaussian noise with unknown covariance. The original hypothesis test problem is formulated as the discrimination between distributions of the measurements and the noise. The Siegel distance, which is exactly the well-known geodesic distance between images of the original distributions via embedding into a higher-dimensional manifold, is given as an intrinsic measure on the difference between multivariate normal distributions. Without the assumption of uncorrelated measurements, we propose a set of geometric distance detectors, which is designed based on the Siegel distance and different from the generalized likelihood ratio algorithm or other common criterions in statistics. As special cases, the classical optimal matched filter, Rao test, and Wald test, which have been proven to have the CFAR property, belong to the set. Moreover, it is also accessible to an intuitively geometric analysis about how strongly the data contradict the null hypothesis.

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29

CRAMPIN, M., and D. J. SAUNDERS. "Fefferman-type metrics and the projective geometry of sprays in two dimensions." Mathematical Proceedings of the Cambridge Philosophical Society 142, no. 3 (May 2007): 509–23. http://dx.doi.org/10.1017/s0305004107000047.

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AbstractA spray is a second-order differential equation field on the slit tangent bundle of a differentiable manifold, which is homogeneous of degree 1 in the fibre coordinates in an appropriate sense; two sprays which are projectively equivalent have the same base-integral curves up to reparametrization. We show how, when the base manifold is two-dimensional, to construct from any projective equivalence class of sprays a conformal class of metrics on a four-dimensional manifold, of signature (2, 2); the Weyl conformal curvature of these metrics is simply related to the projective curvature of the sprays, and the geodesics of the sprays determine null geodesics of the metrics. The metrics in question have previously been obtained by Nurowski and Sparling (Classical and Quantum Gravity20 (2003) 4995–5016), by a different method involving the exploitation of a close analogy between the Cartan geometry of second-order ordinary differential equations and of three-dimensional Cauchy–Riemann structures. From this perspective the derived metrics are seen to be analoguous to those defined by Fefferman in the CR theory, and are therefore said to be of Fefferman type. Our version of the construction reveals that the Fefferman-type metrics are derivable from the scalar triple product, both directly in the flat case (which we discuss in some detail) and by a simple extension in general. There is accordingly in our formulation a very simple expression for a representative metric of the class in suitable coordinates.

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30

CRANE, LOUIS. "RELATIONAL SPACETIME, MODEL CATEGORIES AND QUANTUM GRAVITY." International Journal of Modern Physics A 24, no. 15 (June 20, 2009): 2753–75. http://dx.doi.org/10.1142/s0217751x0904614x.

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We propose a mathematically concrete way of modelling the suggestion that in quantum gravity the spacetime manifold disappears. We replace the underlying point set topological space with several apparently different models, which are actually related by pairs of adjoint functors from rational homotopy theory. One is a discrete approximation to the causal null path space derived from the multiple images in the spacetime theory of gravitational lensing, described as an object in the model category of differential graded Lie algebras. Another of our models appears as a thickening of spacetime, which we interpret as a formulation of relational geometry. This model is produced from the finite dimensional differential graded algebra of differential forms which can be transmitted out of a finite region consistent with the Bekenstein bound by another functor, called geometric realisation. The thickening of spacetime, which we propose as a version of relational spacetime, has a surprizingly rich structure. Information which would make up a spin bundle over spacetime is contained in it, making it possible to include fermionic fields in a geometric state sum over it. Avenues toward constructing an actual quantum theory of gravity on our models are given a preliminary exploration.

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31

Giansiracusa, Jeffrey. "The Circle Transfer and Cobordism Categories." Proceedings of the Edinburgh Mathematical Society 62, no. 3 (January 11, 2019): 719–31. http://dx.doi.org/10.1017/s0013091518000615.

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AbstractThe circle transfer $Q\Sigma (LX_{hS^1})_+ \to QLX_+$ has appeared in several contexts in topology. In this note, we observe that this map admits a geometric re-interpretation as a morphism of cobordism categories of 0-manifolds and 1-cobordisms. Let 𝒞1(X) denote the one-dimensional cobordism category and let Circ(X) ⊂ 𝒞1(X) denote the subcategory whose objects are disjoint unions of unparametrized circles. Multiplication in S1 induces a functor Circ(X) → Circ(LX), and the composition of this functor with the inclusion of Circ(LX) into 𝒞1(LX) is homotopic to the circle transfer. As a corollary, we describe the inclusion of the subcategory of cylinders into the two-dimensional cobordism category 𝒞2(X) and find that it is null-homotopic when X is a point.

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32

Mirzaiyan, Zahra, and Giampiero Esposito. "On the Nature of Bondi–Metzner–Sachs Transformations." Symmetry 15, no. 4 (April 21, 2023): 947. http://dx.doi.org/10.3390/sym15040947.

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This paper investigates, as a first step, the four branches of BMS transformations, motivated by the classification into elliptic, parabolic, hyperbolic and loxodromic proposed a few years ago in the literature. We first prove that to each normal elliptic transformation of the complex variable ζ used in the metric for cuts of null infinity, there is a corresponding BMS supertranslation. We then study the conformal factor in the BMS transformation of the u variable as a function of the squared modulus of ζ. In the loxodromic and hyperbolic cases, this conformal factor is either monotonically increasing or monotonically decreasing as a function of the real variable given by the modulus of ζ. The Killing vector field of the Bondi metric is also studied in correspondence with the four admissible families of BMS transformations. Eventually, all BMS transformations are re-expressed in the homogeneous coordinates suggested by projective geometry. It is then found that BMS transformations are the restriction to a pair of unit circles of a more general set of transformations. Within this broader framework, the geometry of such transformations is studied by means of its Segre manifold.

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33

Palomo, Francisco J. "The fibre bundle of degenerate tangent planes of a Lorentzian manifold and the smoothness of the null sectional curvature." Differential Geometry and its Applications 25, no. 6 (December 2007): 667–73. http://dx.doi.org/10.1016/j.difgeo.2007.06.013.

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34

ETAYO, FERNANDO, ARACELI DEFRANCISCO, and RAFAEL SANTAMARÍA. "Classification of pure metallic metric geometries." Carpathian Journal of Mathematics 38, no. 2 (February 28, 2022): 417–29. http://dx.doi.org/10.37193/cjm.2022.02.12.

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Metallic Riemannian manifolds with null trace and metallic Norden manifolds are generalizations of almost product Riemannian and almost golden Riemannian manifolds with null trace and almost Norden and almost Norden golden manifolds respectively. All these pure metrics geometries can be unified under the notion of α-metallic metric manifold. We classify this kind of manifolds in a consistent way with the well-known classifications of almost product Riemannian manifolds with null trace and almost Norden manifolds. We also characterize all classes of α-metallic metric manifolds by means of the first canonical connection which is a distinguished adapted connection.

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35

Noguchi, Mitsunori. "Geometry of statistical manifolds." Differential Geometry and its Applications 2, no. 3 (September 1992): 197–222. http://dx.doi.org/10.1016/0926-2245(92)90011-b.

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36

Aspinwall, P. S., and C. A. Lütken. "Geometry of mirror manifolds." Nuclear Physics B 353, no. 2 (April 1991): 427–61. http://dx.doi.org/10.1016/0550-3213(91)90343-v.

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37

LIU, KE-FENG, and XIAO-KUI YANG. "GEOMETRY OF HERMITIAN MANIFOLDS." International Journal of Mathematics 23, no. 06 (May 6, 2012): 1250055. http://dx.doi.org/10.1142/s0129167x12500553.

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On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (and Riemannian real vector bundle) with an arbitrary metric connection over a compact Hermitian manifold, we can derive various vanishing theorems for Hermitian manifolds and complex vector bundles by the second Ricci curvature tensors. We will also introduce a natural geometric flow on Hermitian manifolds by using the second Ricci curvature tensor.

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38

ARVANITOYEORGOS, ANDREAS. "GEOMETRY OF FLAG MANIFOLDS." International Journal of Geometric Methods in Modern Physics 03, no. 05n06 (September 2006): 957–74. http://dx.doi.org/10.1142/s0219887806001399.

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A flag manifold is a homogeneous space M = G/K, where G is a compact semisimple Lie group, and K the centralizer of a torus in G. Equivalently, M can be identified with the adjoint orbit Ad (G)w of an element w in the Lie algebra of G. We present several aspects of flag manifolds, such as their classification in terms of painted Dynkin diagrams, T-roots and G-invariant metrics, and Kähler metrics. We give a Lie-theoretic expression of the Ricci tensor in M, hence reducing the Einstein equation on flag manifolds into an algebraic system of equations, which can be solved in several cases. A flag manifold is also a complex manifold, and this dual representation as a real and a complex manifold is related to a similar property of an infinite-dimensional manifold, the loop space, which in fact can be viewed as a "universal" flag manifold.

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39

Rosly, A. A., A. S. Schwarz, and A. A. Voronov. "Geometry of superconformal manifolds." Communications in Mathematical Physics 119, no. 1 (March 1988): 129–52. http://dx.doi.org/10.1007/bf01218264.

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40

Shima, Hirohiko, and Katsumi Yagi. "Geometry of Hessian manifolds." Differential Geometry and its Applications 7, no. 3 (September 1997): 277–90. http://dx.doi.org/10.1016/s0926-2245(96)00057-5.

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41

Voronov, Theodore Th. "Graded Geometry, Q ‐Manifolds, and Microformal Geometry." Fortschritte der Physik 67, no. 8-9 (May 6, 2019): 1910023. http://dx.doi.org/10.1002/prop.201910023.

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42

Ssekajja, Samuel. "On contact screen conformal null submanifolds." Boletim da Sociedade Paranaense de Matemática 41 (December 24, 2022): 1–8. http://dx.doi.org/10.5269/bspm.51992.

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First, we prove that indefinite Sasakian manifolds do not admit any screen conformal $r$-null submanifolds, tangent to the structure vector field. We, therefore, define a special class of null submanifolds, called; {\it contact screen conformal} $r$-null submanifold of indefinite Sasakian manifolds. Several characterization results, on the above class of null submanifolds, are proved. In particular, we prove that such null submanifolds exist in indefinite Sasakian space forms of constant holomorphic sectional curvatures of $-3$.

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43

MATSUZOE, Hiroshi. "GEOMETRY OF SEMI-WEYL MANIFOLDS AND WEYL MANIFOLDS." Kyushu Journal of Mathematics 55, no. 1 (2001): 107–17. http://dx.doi.org/10.2206/kyushujm.55.107.

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YASUHARA, AKIRA. "NULL-HOMOLOGOUS LINKS IN CERTAIN 4-MANIFOLDS." Journal of Knot Theory and Its Ramifications 08, no. 01 (February 1999): 115–23. http://dx.doi.org/10.1142/s0218216599000092.

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Let M be a 4-manifold with ∂M ≅ S3 and L⊂ ∂M a link. The link L is null-homologous in M if L bounds a disjoint union of once-punctured, orientable surfaces in M. In a previous paper [1] the author defined null-homologous link in 4-manifolds and gave a necessary and sufficient condition for links to be null-homologous in 4-manifolds. By using this condition, we investigate the sets of null-homologous links in punctured [Formula: see text], [Formula: see text], [Formula: see text] and S2 × S2.

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45

Seahra, Sanjeev S., and Paul S. Wesson. "Null Geodesics in Five-Dimensional Manifolds." General Relativity and Gravitation 33, no. 10 (October 2001): 1731–52. http://dx.doi.org/10.1023/a:1013023100565.

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46

Bahn, Hyoungsick, Young Do Chai, and Sungpyo Hong. "Infinitesimally null Ricci isotropic Lorentz manifolds." Journal of Geometry and Physics 20, no. 2-3 (October 1996): 297–300. http://dx.doi.org/10.1016/0393-0440(95)00055-0.

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47

BOYER, CHARLES P., and KRZYSZTOF GALICKI. "ON SASAKIAN–EINSTEIN GEOMETRY." International Journal of Mathematics 11, no. 07 (September 2000): 873–909. http://dx.doi.org/10.1142/s0129167x00000477.

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We introduce a multiplication ⋆ (we call it a join) on the space of all compact Sasakian-Einstein orbifolds [Formula: see text] and show that [Formula: see text] has the structure of a commutative associative topological monoid. The set [Formula: see text] of all compact regular Sasakian–Einstein manifolds is then a submonoid. The set of smooth manifolds in [Formula: see text] is not closed under this multiplication; however, the join [Formula: see text] of two Sasakian–Einstein manifolds is smooth under some additional conditions which we specify. We use this construction to obtain many old and new examples of Sasakain–Einstein manifolds. In particular, in every odd dimension greater that five we obtain spaces with arbitrary second Betti number.

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Rustanov, Aligadzhi R. "Geometry of Harmonic Nearly Trans-Sasakian Manifolds." Axioms 12, no. 8 (July 28, 2023): 744. http://dx.doi.org/10.3390/axioms12080744.

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This paper considers a class of nearly trans-Sasakian manifolds. The local structure of nearly trans-Sasakian structures with a closed contact form and a closed Lee form is obtained. It is proved that the class of nearly trans-Sasakian manifolds with a closed contact form and a closed Lee form coincides with the class of almost contact metric manifolds with a closed contact form locally conformal to the closely cosymplectic manifolds. A wide class of harmonic nearly trans-Sasakian manifolds has been identified (i.e., nearly trans-Sasakian manifolds with a harmonic contact form) and an exhaustive description of the manifolds of this class is obtained. Also, examples of harmonic nearly trans-Sasakian manifolds are given.

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T. Venkatesh, Et al. "Group Actions on Manifolds." Communications on Applied Nonlinear Analysis 31, no. 1 (January 31, 2024): 136–40. http://dx.doi.org/10.52783/cana.v31.357.

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In this paper some Riemann Geometry aspects will be gathered with classical time, the study naturally concentrate to PDE’s from the relation outside geometry. The classical frame work of differential geometry (intending smooth manifolds) and later it endured with Riemannian metric is briefly described in one of the sections that were formed to be essential for our understanding of the inner structure of the space.

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50

Tyurin, N. A. "Lagrangian Geometry of Algebraic Manifolds." Physics of Particles and Nuclei Letters 19, no. 4 (July 26, 2022): 337–42. http://dx.doi.org/10.1134/s1547477122040215.

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