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Journal articles on the topic 'Hermitian metric'

Author: Grafiati

Published: 6 September 2023

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1

Kawamura, Masaya. "On Kähler-like and G-Kähler-like almost Hermitian manifolds." Complex Manifolds 7, no. 1 (April 3, 2020): 145–61. http://dx.doi.org/10.1515/coma-2020-0009.

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AbstractWe introduce Kähler-like, G-Kähler-like metrics on almost Hermitian manifolds. We prove that a compact Kähler-like and G-Kähler-like almost Hermitian manifold equipped with an almost balanced metric is Kähler. We also show that if a Kähler-like and G-Kähler-like almost Hermitian manifold satisfies B_{\bar i\bar j}^\lambda B_{\lambda j}^i \ge 0, then the metric is almost balanced and the almost complex structure is integrable, which means that the metric is balanced. We investigate a G-Kähler-like almost Hermitian manifold under some assumptions.

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2

Faran, V, James J. "Hermitian Finsler metrics and the Kobayashi metric." Journal of Differential Geometry 31, no. 3 (1990): 601–25. http://dx.doi.org/10.4310/jdg/1214444630.

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3

ALDEA, NICOLETA, and GHEORGHE MUNTEANU. "NEW CANDIDATES FOR A HERMITIAN APPROACH OF GRAVITY." International Journal of Geometric Methods in Modern Physics 10, no. 09 (August 30, 2013): 1350041. http://dx.doi.org/10.1142/s0219887813500412.

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In this paper, some possible candidates for the study of gravity are proposed in terms of complex Finsler geometry. These mainly concern the complex Hermitian versions of weakly gravitational metric and Schwarzschild metric. For the weakly gravitational fields, we state few interesting geometrical and physical aspects such as the conditions under which a complex Finsler metrics are projectively related to the weakly gravitational metric. In the Kähler case, the geodesic curves of the weakly gravitational metric are obtained. Some applications concerning the deformations of the weakly gravitational Hermitian metric to a complex Randers metric are described. Another candidate for gravity is given by so-called Hermitian Schwarzschild metric for which some geodesic curves are highlighted. The last part of the paper is devoted to a generalization of the complex Klein–Gordon equations, in terms of Quantum field theory on a curved space.

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4

Talmadge, Andrew. "Symmetry breaking via internal geometry." International Journal of Mathematics and Mathematical Sciences 2005, no. 13 (2005): 2023–30. http://dx.doi.org/10.1155/ijmms.2005.2023.

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Gauge theories commonly employ complex vector-valued fields to reduce symmetry groups through the Higgs mechanism of spontaneous symmetry breaking. The geometry of the internal spaceVis tacitly assumed to be the metric geometry of some static, nondynamical hermitian metrick. In this paper, we considerG-principal bundle gauge theories, whereGis a subgroup ofU(V,k)(the unitary transformations on the internal vector spaceVwith hermitian metrick) and we consider allowing the hermitian metric on the internal spaceVto become an additional dynamical element of the theory. We find a mechanism for interpreting the Higgs scalar field as a feature of the geometry of the internal space while retaining the successful aspects of the Higgs mechanism and spontaneous symmetry breaking

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5

Zelewski, Piotr M. "On the Hermitian-Einstein Tensor of a Complex Homogenous Vector Bundle." Canadian Journal of Mathematics 45, no. 3 (June 1, 1993): 662–72. http://dx.doi.org/10.4153/cjm-1993-037-5.

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AbstractWe prove that any holomorphic, homogenous vector bundle admits a homogenous minimal metric—a metric for which the Hermitian-Einstein tensor is diagonal in a suitable sense. The concept of minimality depends on the choice of the Jordan-Holder filtration of the corresponding parabolic module. We show that the set of all admissible Hermitian-Einstein tensors of certain class of minimal metrics is a convex subset of the euclidean space. As an application, we obtain an algebraic criterion for semistability of homogenous holomorphic vector bundles.

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6

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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7

Salimov, Arif. "On anti-Hermitian metric connections." Comptes Rendus Mathematique 352, no. 9 (September 2014): 731–35. http://dx.doi.org/10.1016/j.crma.2014.07.004.

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8

Mostafazadeh, Ali. "Pseudo-Hermitian quantum mechanics with unbounded metric operators." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1989 (April 28, 2013): 20120050. http://dx.doi.org/10.1098/rsta.2012.0050.

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I extend the formulation of pseudo-Hermitian quantum mechanics to η + -pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator η + . In particular, I give the details of the construction of the physical Hilbert space, observables and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of η + and consequently .

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9

,, Haripamyu, Jenizon ,, and I. Made Arnawa. "Metrik Finsler Pseudo-Konveks Kuat pada Bundel Vektor Holomorfik." Jurnal Matematika 7, no. 1 (June 10, 2017): 12. http://dx.doi.org/10.24843/jmat.2017.v07.i01.p78.

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Abstract: Rizza-negativity of holomorphic vector bundle is a sufficient condition for the negativity of . In the present paper, we shall discuss that as a special case, using the Rizza metric which is derived from a Hermitian metric also implies the negativity of . Further we showed that for the negative holomorphic vector bundle there is a pseudo-convex Finsler metric with negative curvature. Keywords: Hermitian metric, Rizza metric, Rizza-negativity.

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10

VERGARA-DIAZ, E., and C. M. WOOD. "HARMONIC CONTACT METRIC STRUCTURES AND SUBMERSIONS." International Journal of Mathematics 20, no. 02 (February 2009): 209–25. http://dx.doi.org/10.1142/s0129167x09005224.

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We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby–Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonicity to that of the almost Hermitian structure on the base or fibre.

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11

Biswas, Indranil, and Georg Schumacher. "Deligne pairing and Quillen metric." International Journal of Mathematics 25, no. 14 (December 2014): 1450122. http://dx.doi.org/10.1142/s0129167x14501225.

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Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over ℂ. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing 〈L,…,L〉 → S with the determinant line bundle [Formula: see text] (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen–Mumford expansion, J. Differential Geom. 78 (2008) 475–496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles [Formula: see text] and 〈L,…,L〉 carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between 〈L,…,L〉 → S and [Formula: see text] is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91–96] between a Deligne paring and a certain determinant line bundle.

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12

Banaru, G. A. "On quasi-Sasakian structure on a totally umbilical hypersurface of a six-dimensional Hermitian planar submanifold of Cayley algebra." Differential Geometry of Manifolds of Figures, no. 53 (2022): 5–12. http://dx.doi.org/10.5922/0321-4796-2022-53-1.

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Six-dimensional planar submanifolds of Cayley algebra equipped with almost Hermitian structures induced by Brown — Gray three-fold vector cross products in are considered. We select the case when the almost Hermitian structures on such six-dimensional planar submanifolds of Cayley algebra are Hermitian, i. e. these structures are integrable. We study almost contact metric structures on totally umbilical hypersurfaces in such six-dimensional Hermitian planar submanifolds of the octave algebra. We prove that if these almost contact metric structures on a totally umbilical hypersurface of a six-dimensional Hermitian planar submanifold of Cayley algebra are quasi-Sasakian, then they are Sasakian.

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13

STEMMLER, MATTHIAS. "STABILITY AND HERMITIAN–EINSTEIN METRICS FOR VECTOR BUNDLES ON FRAMED MANIFOLDS." International Journal of Mathematics 23, no. 09 (July 31, 2012): 1250091. http://dx.doi.org/10.1142/s0129167x12500917.

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We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford–Takemoto and Hermitian–Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that KX ⊗ [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization KX ⊗ [D] coincides with the degree with respect to the complete Kähler–Einstein metric gX\D on X\D. For stable holomorphic vector bundles, we prove the existence of a Hermitian–Einstein metric with respect to gX\D and also the uniqueness in an adapted sense.

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14

Pan, Chung-Ming. "Singular Gauduchon metrics." Compositio Mathematica 158, no. 6 (June 2022): 1314–28. http://dx.doi.org/10.1112/s0010437x22007618.

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In 1977, Gauduchon proved that on every compact hermitian manifold $(X, \omega )$ there exists a conformally equivalent hermitian metric $\omega _\mathrm {G}$ which satisfies $\mathrm {dd}^{\mathrm {c}} \omega _\mathrm {G}^{n-1} = 0$ . In this note, we extend this result to irreducible compact singular hermitian varieties which admit a smoothing.

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15

Ali, Danish, Johann Davidov, and Oleg Mushkarov. "Holomorphic curvatures of twistor spaces." International Journal of Geometric Methods in Modern Physics 11, no. 03 (March 2014): 1450022. http://dx.doi.org/10.1142/s0219887814500224.

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We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on ℂℙ3 is a non-Kähler Hermitian–Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein–Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z.190 (1985) 39–43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom.7 (2007) 207–214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

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16

Kawamura, Masaya. "On the Kähler-likeness on almost Hermitian manifolds." Complex Manifolds 6, no. 1 (January 1, 2019): 366–76. http://dx.doi.org/10.1515/coma-2019-0020.

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AbstractWe define a Kähler-like almost Hermitian metric. We will prove that on a compact Kähler-like almost Hermitian manifold (M2n, J, g), if it admits a positive ∂ ̄∂-closed (n − 2, n − 2)-form, then g is a quasi-Kähler metric.

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17

Cho, Jong Taek. "Contact Metric Spaces and pseudo-Hermitian Symmetry." Mathematics 8, no. 9 (September 14, 2020): 1583. http://dx.doi.org/10.3390/math8091583.

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We prove that a contact strongly pseudo-convex CR (Cauchy–Riemann) manifold M2n+1, n≥2, is locally pseudo-Hermitian symmetric and satisfies ∇ξh=μhϕ, μ∈R, if and only if M is either a Sasakian locally ϕ-symmetric space or a non-Sasakian (k,μ)-space. When n=1, we prove a classification theorem of contact strongly pseudo-convex CR manifolds with pseudo-Hermitian symmetry.

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18

La Cruz, Javier De, Jorge Robinson Evilla, and Ferruh Ozbudak. "Hermitian Rank Metric Codes and Duality." IEEE Access 9 (2021): 38479–87. http://dx.doi.org/10.1109/access.2021.3064503.

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19

Ohmori, S., and J. Takahashi. "Rigged Hilbert space approach for non-Hermitian systems with positive definite metric." Journal of Mathematical Physics 63, no. 12 (December 1, 2022): 123503. http://dx.doi.org/10.1063/5.0123947.

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We investigate Dirac’s bra–ket formalism based on a rigged Hilbert space for a non-Hermitian quantum system with a positive-definite metric. First, the rigged Hilbert space, characterized by a positive-definite metric, is established. With the aid of the nuclear spectral theorem for the obtained rigged Hilbert space, spectral expansions are shown for the bra–kets by the generalized eigenvectors of a quasi-Hermitian operator. The spectral expansions are utilized to endow the complete bi-orthogonal system and the transformation theory between the Hermitian and non-Hermitian systems. As an example of application, we show a specific description of our rigged Hilbert space treatment for some parity-time symmetrical quantum systems.

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20

JAKUBSKÝ, VÍT. "THERMODYNAMICS OF PSEUDO-HERMITIAN SYSTEMS IN EQUILIBRIUM." Modern Physics Letters A 22, no. 15 (May 20, 2007): 1075–84. http://dx.doi.org/10.1142/s0217732307023419.

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In the study of pseudo(quasi)-hermitian operators, the key role is played by the positive-definite metric operator. It enables physical interpretation of the considered systems. In this paper, we study the pseudo-hermitian systems with constant number of particles in equilibrium. We show that the explicit knowledge of the metric operator is not essential for the study of thermodynamic properties of the system. We introduce a simple example where the physically relevant quantities are derived without explicit calculation of either metric operator or spectrum of the Hamiltonian.

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21

Yeşiltaş, Özlem. "Non-Hermitian Dirac Hamiltonian in Three-Dimensional Gravity and Pseudosupersymmetry." Advances in High Energy Physics 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/484151.

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The Dirac Hamiltonian in the(2+1)-dimensional curved space-time has been studied with a metric for an expanding de Sitter space-time which is two spheres. The spectrum and the exact solutions of the time dependent non-Hermitian and angle dependent Hamiltonians are obtained in terms of the Jacobi and Romanovski polynomials. Hermitian equivalent of the Hamiltonian obtained from the Dirac equation is discussed in the frame of pseudo-Hermiticity. Furthermore, pseudosupersymmetric quantum mechanical techniques are expanded to a curved Dirac Hamiltonian and a partner curved Dirac Hamiltonian is generated. Usingη-pseudo-Hermiticity, the intertwining operator connecting the non-Hermitian Hamiltonians to the Hermitian counterparts is found. We have obtained a new metric tensor related to the new Hamiltonian.

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22

RIVERS, R. J. "PATH INTEGRALS FOR QUASI-HERMITIAN HAMILTONIANS." International Journal of Modern Physics D 20, no. 05 (May 20, 2011): 919–32. http://dx.doi.org/10.1142/s0218271811019128.

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Inner products in quasi-Hermitian quantum theories, and hence probabilities, are defined through a metric that depends on the details of the Hamiltonians themselves. We shall see that the functional integral for quasi-Hermitian theories, and hence Feynman diagrams, for example, can be calculated without needing to evaluate the metric. The reason turns out be that their derivation is based fundamentally on the Heisenberg equations of motion and the canonical equal-time commutation relations, which retain their standard form. As an application, we show how co-ordinate transformations in the path integral can enable us to recover equivalent Hermitian Hamiltonians.

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23

JANYŠKA, JOSEF, and MARCO MODUGNO. "HERMITIAN VECTOR FIELDS AND SPECIAL PHASE FUNCTIONS." International Journal of Geometric Methods in Modern Physics 03, no. 04 (June 2006): 719–54. http://dx.doi.org/10.1142/s0219887806001351.

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We start by analyzing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.

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24

Cirici, Joana, and Scott O. Wilson. "Almost Hermitian Identities." Mathematics 8, no. 8 (August 13, 2020): 1357. http://dx.doi.org/10.3390/math8081357.

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We study the local commutation relation between the Lefschetz operator and the exterior differential on an almost complex manifold with a compatible metric. The identity that we obtain generalizes the backbone of the local Kähler identities to the setting of almost Hermitian manifolds, allowing for new global results for such manifolds.

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25

Vitter, Al. "Hermitian–Einstein metrics and jumping lines forSp(n+1)-homogeneous bundles over ℙ2n+1." Mathematical Proceedings of the Cambridge Philosophical Society 114, no. 3 (November 1993): 443–51. http://dx.doi.org/10.1017/s0305004100071735.

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Stable holomorphic vector bundles over complex projective space ℙnhave been studied from both the differential-geometric and the algebraic-geometric points of view.On the differential-geometric side, the stability ofE-→ ℙncan be characterized by the existence of a unique hermitian–Einstein metric onE, i.e. a metric whose curvature matrix has trace-free part orthogonal to the Fubini–Study Kähler form of ℙn(see [6], [7], and [13]). Very little is known about this metric in general and the only explicit examples are the metrics on the tangent bundle of ℙnand the nullcorrelation bundle (see [9] and [10]).

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26

Piovani, Riccardo, and Tommaso Sferruzza. "Deformations of Strong Kähler with torsion metrics." Complex Manifolds 8, no. 1 (January 1, 2021): 286–301. http://dx.doi.org/10.1515/coma-2020-0120.

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Abstract Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics {ω t } t which equals a fixed SKT metric ω for t = 0, along a differentiable family of complex manifolds {M t } t .

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27

Lee, John M. "The Fefferman metric and pseudo-Hermitian invariants." Transactions of the American Mathematical Society 296, no. 1 (January 1, 1986): 411. http://dx.doi.org/10.1090/s0002-9947-1986-0837820-2.

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28

Goldberg, Bryan, and Rongwei Yang. "Hermitian Metric and the Infinite Dihedral Group." Proceedings of the Steklov Institute of Mathematics 304, no. 1 (January 2019): 136–45. http://dx.doi.org/10.1134/s0081543819010097.

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29

Quiroga, R. "Hermitian metric rigidity on compact foliated manifolds." Geometriae Dedicata 57, no. 3 (October 1995): 305–15. http://dx.doi.org/10.1007/bf01263487.

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30

Khan, Mohammad Nazrul Islam. "TANGENT BUNDLE ENDOWED WITH QUARTER-SYMMETRIC NON-METRIC CONNECTION ON AN ALMOST HERMITIAN MANIFOLD." Facta Universitatis, Series: Mathematics and Informatics 35, no. 1 (April 6, 2020): 167. http://dx.doi.org/10.22190/fumi2001167k.

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In this paper, we have studied the tangent bundle endowed with quarter-symmetric non-metric connection obtained by vertical and complete lifts of a quarter-symmetric non-metric connection on the base manifold and, also, proposed the study of the tangent bundle of an almost Hermitian manifold and an almost Kaehler manifold. Finally, we obtained some theorems for Nijenhuis tensor on the tangent bundle of an almost Hermitian manifold and an almost Kaehler manifold.\\

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31

Banaru, G. A. "On nearly Kählerian manifolds and quasi-Sasakian hypersurfaces axiom." Differential Geometry of Manifolds of Figures, no. 52 (2021): 17–22. http://dx.doi.org/10.5922/0321-4796-2021-52-2.

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It is known that an almost contact metric structure is induced on an arbitrary hypersurface of an almost Hermitian manifold. The case when the almost Hermitian manifold is nearly Kählerian and the almost contact metric structure on its hypersurface is quasi-Sasakian is considered. It is proved that non-Kählerian nearly Kählerian manifolds (in particular, the six-dimensional sphere equipped with the canonical nearly Kählerian structure) do not satisfy to the quasi-Sasakian hypersurfaces axiom.

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32

Mostafazadeh, Ali. "Metric operators for quasi-Hermitian Hamiltonians and symmetries of equivalent Hermitian Hamiltonians." Journal of Physics A: Mathematical and Theoretical 41, no. 24 (June 3, 2008): 244017. http://dx.doi.org/10.1088/1751-8113/41/24/244017.

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33

Cardona, Sergio A. H., and Claudio Meneses. "On a functional of Kobayashi for Higgs bundles." International Journal of Geometric Methods in Modern Physics 17, no. 13 (October 12, 2020): 2050200. http://dx.doi.org/10.1142/s021988782050200x.

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We define a functional [Formula: see text] for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that [Formula: see text] is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite–Yang–Mills metrics. We derive a formula relating [Formula: see text] and another functional [Formula: see text], closely related to the Yang–Mills–Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of [Formula: see text], which is expressed as a certain [Formula: see text]-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of [Formula: see text] if and only if the corresponding Hitchin–Simpson mean curvature is parallel with respect to the Hitchin–Simpson connection.

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34

Rustanov, Aligadzhi. "Nearly Sasakian Manifolds of Constant Type." Axioms 11, no. 12 (November 26, 2022): 673. http://dx.doi.org/10.3390/axioms11120673.

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The article deals with nearly Sasakian manifolds of a constant type. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the nearly Sasakian manifold is a nearly Kähler manifold. It is proved that the class of nearly Sasakian manifolds of the zero constant type coincides with the class of Sasakian manifolds. The concept of constancy of the type of an almost contact metric manifold is introduced through its Nijenhuis tensor, and the criterion of constancy of the type of an almost contact metric manifold is proved. The coincidence of both concepts of type constancy for the nearly Sasakian manifold is proved. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the almost contact metric manifold of the zero constant type is the Hermitian structure.

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35

DAVIS, ALFRED, and TRISTAN HÜBSCH. "A FERMIONIC HODGE STAR OPERATOR." Modern Physics Letters A 14, no. 15 (May 20, 1999): 965–76. http://dx.doi.org/10.1142/s0217732399001036.

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A fermionic analogue of the Hodge star operation is shown to have an explicit operator representation in models with fermions, in space–times of any dimension. This operator realizes a conjugation (pairing) not used explicitly in field theory, and induces a metric in the space of wave function(al)s just as in exterior calculus. If made real (hermitian), this induced metric turns out to be identical to the standard one constructed using hermitian conjugation; the utility of the induced complex bilinear form remains unclear.

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36

Biswas, Indranil, and Vamsi Pritham Pingali. "Metric Properties of Parabolic Ample Bundles." International Mathematics Research Notices 2020, no. 23 (November 5, 2018): 9336–69. http://dx.doi.org/10.1093/imrn/rny259.

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Abstract We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern–Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favor for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type result holds and that there are metrics on stable bundles over surfaces whose Schur forms are positive.

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37

Abbena, Elsa, and Sergio Garbiero. "Almost Hermitian homogeneous structures." Proceedings of the Edinburgh Mathematical Society 31, no. 3 (October 1988): 375–95. http://dx.doi.org/10.1017/s0013091500006775.

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Let (M, g, J) be an almost Hermitian manifold. More precisely, M is a ∞ differentiable manifold of dimension 2n, J is an almost complex structure on M, i.e. it is a tensor field of type (1, 1) such thatfor any X∈(M), ((M) is the Lie algebra of ∞ vector fields on M), and g is a Riemannian metric compatible with J, i.e.

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38

Yang, Kichoon. "Invariant Kähler metrics and projective embeddings of the flag manifold." Bulletin of the Australian Mathematical Society 49, no. 2 (April 1994): 239–47. http://dx.doi.org/10.1017/s0004972700016300.

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We determine explicitly the space of invariant Hermitian and Kähler metrics on the flag manifold. In particular, we show that a Killing metric is not Kähler. The Chern forms are also computed in terms of the Maurer–Cartan form, and this calculation is used to prove that the flag manifold is projective algebraic. An explicit projective embedding of the flag manifold is also given.

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39

Assis, Paulo E. G. "Metric operators for non-Hermitian quadraticsu(2) Hamiltonians." Journal of Physics A: Mathematical and Theoretical 44, no. 26 (June 1, 2011): 265303. http://dx.doi.org/10.1088/1751-8113/44/26/265303.

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40

Mok, N. "Metric rigidity theorems on Hermitian locally symmetric spaces." Proceedings of the National Academy of Sciences 83, no. 8 (April 1, 1986): 2288–90. http://dx.doi.org/10.1073/pnas.83.8.2288.

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41

Ergun, Ebru, and Mesude Saglam. "On the metric of a non-Hermitian model." Reports on Mathematical Physics 65, no. 3 (June 2010): 367–78. http://dx.doi.org/10.1016/s0034-4877(10)00015-7.

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SHALABY, ABOUZEID M. "REPRESENTATION DEPENDENCE OF SUPERFICIAL DEGREE OF DIVERGENCES IN QUANTUM FIELD THEORY." International Journal of Modern Physics A 26, no. 17 (July 10, 2011): 2913–25. http://dx.doi.org/10.1142/s0217751x1105364x.

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In this work, we investigate a very important but unstressed result in the work of C. M. Bender, J.-H. Chen, and K. A. Milton, J. Phys. A39, 1657 (2006). These authors have calculated the vacuum energy of the iϕ3 scalar field theory and its Hermitian equivalent theory up to g4 order of calculations. While all the Feynman diagrams of the iϕ3 theory are finite in 0+1 space–time dimensions, some of the corresponding Feynman diagrams in the equivalent Hermitian theory are divergent. In this work, we show that the divergences in the Hermitian theory originate from superrenormalizable, renormalizable and nonrenormalizable terms in the interaction Hamiltonian even though the calculations are carried out in the 0+1 space–time dimensions. Relying on this interesting result, we raise a question: Is the superficial degree of divergence of a theory is representation dependent? To answer this question, we introduce and study a class of non-Hermitian quantum field theories characterized by a field derivative interaction Hamiltonian. We showed that the class is physically acceptable by finding the corresponding class of metric operators in a closed form. We realized that the obtained equivalent Hermitian and the introduced non-Hermitian representations have coupling constants of different mass dimensions which may be considered as a clue for the possibility of considering nonrenormalizability of a field theory as a nongenuine problem. Besides, the metric operator is supposed to disappear from path integral calculations which means that physical amplitudes can be fully obtained in the simpler non-Hermitian representation.

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43

Balcerzak, Bogdan. "On Symmetric Brackets Induced by Linear Connections." Symmetry 13, no. 6 (June 3, 2021): 1003. http://dx.doi.org/10.3390/sym13061003.

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In this note, we discuss symmetric brackets on skew-symmetric algebroids associated with metric or symplectic structures. Given a pseudo-Riemannian metric structure, we describe the symmetric brackets induced by connections with totally skew-symmetric torsion in the language of Lie derivatives and differentials of functions. We formulate a generalization of the fundamental theorem of Riemannian geometry. In particular, we obtain an explicit formula of the Levi-Civita connection. We also present some symmetric brackets on almost Hermitian manifolds and discuss the first canonical Hermitian connection. Given a symplectic structure, we describe symplectic connections using symmetric brackets. We define a symmetric bracket of smooth functions on skew-symmetric algebroids with the metric structure and show that it has properties analogous to the Lie bracket of Hamiltonian vector fields on symplectic manifolds.

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MANDAl, BHABANI PRASAD, and SAURABH GUPTA. "PSEUDO-HERMITIAN INTERACTIONS IN DIRAC THEORY: EXAMPLES." Modern Physics Letters A 25, no. 20 (June 28, 2010): 1723–32. http://dx.doi.org/10.1142/s0217732310032901.

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We consider a couple of examples to study the pseudo-Hermitian interaction in relativistic quantum mechanics. Rasbha interaction, commonly used to study the spin Hall effect, is considered with imaginary coupling. The corresponding Dirac Hamiltonian is shown to be parity pseudo-Hermitian. In the other example we consider parity pseudo-Hermitian scalar interaction with arbitrary parameter in Dirac theory. In both cases we show that the energy spectrum is real and all the other features of nonrelativistic pseudo-Hermitian formulation are present. Using the spectral method, the positive definite metric operator (η) has been calculated explicitly for both the models to ensure positive definite norms for the state vectors.

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Feinberg, Joshua, and Roman Riser. "Pseudo-hermitian random matrix theory: a review." Journal of Physics: Conference Series 2038, no. 1 (October 1, 2021): 012009. http://dx.doi.org/10.1088/1742-6596/2038/1/012009.

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Abstract We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of this new type of random matrices, we focus on two specific models of matrices which are pseudo-hermitian with respect to a given indefinite metric B. Eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. The diagrammatic method is applied to deriving explicit analytical expressions for the density of eigenvalues in the complex plane and on the real axis, in the large-N, planar limit. In one of the models we discuss, the metric B depends on a certain real parameter t. As t varies, the model exhibits various ‘phase transitions’ associated with eigenvalues flowing from the complex plane onto the real axis, causing disjoint eigenvalue support intervals to merge. Our analytical results agree well with presented numerical simulations.

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DOBRSKI, MICHAŁ. "INVOLUTION IN QUANTIZED ENDOMORPHISM BUNDLE AND REALITY OF NONCOMMUTATIVE GRAVITY ACTIONS." International Journal of Geometric Methods in Modern Physics 10, no. 02 (December 5, 2012): 1220029. http://dx.doi.org/10.1142/s0219887812200290.

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It is shown that for arbitrary connection in the vector bundle compatible with some Hermitian metric, the corresponding Fedosov trace functional commutes with involution generated by this metric. This result is then used to prove that certain noncommutative gravity actions are real in all powers of deformation parameter.

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Dancer, Andrew S., and Ian A. B. Strachan. "Kähler–Einstein metrics with SU(2) action." Mathematical Proceedings of the Cambridge Philosophical Society 115, no. 3 (May 1994): 513–25. http://dx.doi.org/10.1017/s0305004100072273.

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The aim of this paper is to analyse Riemannian Kähler–Einstein metrics g in real dimension four admitting an isometric action of SU(2) with generically three-dimensional orbits. The Kähler condition means that there is a complex structure I, with respect to which the metric is hermitian, such that the two-form Ωdefined byis closed. It is well-known that if this condition holds then Ω is in fact covariant constant.

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Tshikuna-Matamba, T. "The differential geometry of almost Hermitian almost contact metric submersions." International Journal of Mathematics and Mathematical Sciences 2004, no. 36 (2004): 1923–35. http://dx.doi.org/10.1155/s0161171204303364.

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Three types of Riemannian submersions whose total space is an almost Hermitian almost contact metric manifold are studied. The study is focused on fundamental properties and the transference of structures.

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GUAN, DANIEL. "EXISTENCE OF EXTREMAL METRICS ON ALMOST HOMOGENEOUS MANIFOLDS OF COHOMOGENEITY ONE — III." International Journal of Mathematics 14, no. 03 (May 2003): 259–87. http://dx.doi.org/10.1142/s0129167x03001806.

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In this paper we prove that on certain manifolds Nn with nonnegative first Chern class the existence of extremal metric in a Kähler class is the same as the stability of the Kähler class. We also obtain many new Kähler classes with extremal metrics, in particular, the Kähler-Einstein metrics for Nn with n > 2. We also compare the problem of finding Calabi extremal metrics with the similar problem of finding Hermitian–Einstein metrics on the holomorphic vector bundles. We explain the geodesic stability and found that the stability for the manifold is much more complicated

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Ni, Qihui, Yong He, Jinhua Yang, and Hui Zhang. "Levi-Civita Ricci-Flat Doubly Warped Product Hermitian Manifolds." Advances in Mathematical Physics 2022 (July 5, 2022): 1–7. http://dx.doi.org/10.1155/2022/2077040.

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Let M 1 , g and M 2 , h be two Hermitian manifolds. The doubly warped product (abbreviated as DWP) Hermitian manifold of M 1 , g and M 2 , h is the product manifold M 1 × M 2 endowed with the warped product Hermitian metric G = f 2 2 g + f 1 2 h , where f 1 and f 2 are positive smooth functions on M 1 and M 2 , respectively. In this paper, the formulae of Levi-Civita connection, Levi-Civita curvature, the first Levi-Civita Ricci curvature, and Levi-Civita scalar curvature of the DWP-Hermitian manifold are derived in terms of the corresponding objects of its components. We also prove that if the warped function f 1 and f 2 are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if M 1 , g and M 2 , h are Levi-Civita Ricci-flat manifolds. Thus, we give an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.

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