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Journal articles on the topic 'D-Manifolds'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

GATSE, Servais Cyr. "Jacobi Manifolds, Contact Manifolds and Contactomorphism." Journal of Mathematics Research 13, no. 4 (July 29, 2021): 85. http://dx.doi.org/10.5539/jmr.v13n4p85.

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Let M be a smooth manifold and let D(M) be the module of first order differential operators on M. In this work, we give a link between Jacobi manifolds and Contact manifolds. We also generalize the notion of contactomorphism on M and thus, we characterize the Contact diffeomorphisms.

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2

Rossi, Federico Alberto. "On deformations of D-manifolds and CR D-manifolds." Journal of Geometry and Physics 62, no. 2 (February 2012): 464–78. http://dx.doi.org/10.1016/j.geomphys.2011.11.007.

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3

Ahmad Mirshafeazadeh, Mir, and Behroz Bidabad. "On generalized quasi-Einstein manifolds." Advances in Pure and Applied Mathematics 10, no. 3 (July 1, 2019): 193–202. http://dx.doi.org/10.1515/apam-2017-0112.

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Abstract We study generalized quasi-Einstein manifolds, or briefly, GQE manifolds. Here, we present relations between the Bach, Cotton and D tensors on GQE manifolds. Next, a 3-tensor E which measures the deviation of m-quasi-Einstein manifolds from GQE manifolds is introduced. Among others in dimension 3, it is shown that Bach-flatness implies locally conformally flatness. Furthermore, it is proved that, around a regular point of the fourth-order divergence free Weyl tensor, a GQE manifold is a locally warped product manifold with {(n-1)} -dimensional Einstein fibers in suitable cases.

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4

GRIBACHEVA, DOBRINKA. "A NATURAL CONNECTION ON A BASIC CLASS OF RIEMANNIAN PRODUCT MANIFOLDS." International Journal of Geometric Methods in Modern Physics 09, no. 07 (September 7, 2012): 1250057. http://dx.doi.org/10.1142/s0219887812500570.

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A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

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5

Bershadsky, M., C. Vafa, and V. Sadov. "D-strings on D-manifolds." Nuclear Physics B 463, no. 2-3 (March 1996): 398–414. http://dx.doi.org/10.1016/0550-3213(96)00024-7.

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6

Shah, Riddhi Jung. "Some Curvature Properties of D-conformal Curvature Tensor on LP-Sasakian Manifolds." Journal of Institute of Science and Technology 19, no. 1 (November 8, 2015): 30–34. http://dx.doi.org/10.3126/jist.v19i1.13823.

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This paper deals with the study of geometry of Lorentzian para-Sasakian manifolds. We investigate some properties of D-conformally flat, D-conformally semi-symmetric, Xi-D-conformally flat and Phi-D-conformally flat curvature conditions on Lorentzian para-Sasakian manifolds. Also it is proved that in each curvature condition an LP-Sasakian manifold (Mn,g)(n>3) is an eta-Einstein manifold.Journal of Institute of Science and Technology, 2014, 19(1): 30-34

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7

Śniatycki, Jędrzej. "Generalizations of Frobenius’ Theorem on Manifolds and Subcartesian Spaces." Canadian Mathematical Bulletin 50, no. 3 (September 1, 2007): 447–59. http://dx.doi.org/10.4153/cmb-2007-044-2.

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AbstractLet be a family of vector fields on a manifold or a subcartesian space spanning a distribution D. We prove that an orbit O of is an integral manifold of D if D is involutive on O and it has constant rank on O. This result implies Frobenius’ theorem, and its various generalizations, on manifolds as well as on subcartesian spaces.

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8

Galaev, S. V. "∇N-EINSTEIN ALMOST CONTACT METRIC MANIFOLDS." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 70 (2021): 5–15. http://dx.doi.org/10.17223/19988621/70/1.

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On an almost contact metric manifold M, an N-connection ∇N defined by the pair (∇,N), where ∇ is the interior metric connection and N: TМ → TM is an endomorphism of the tangent bundle of the manifold M such that Nξ = 0, 􀁇 􀁇 N (D) ⊂ D , is considered. Special attention is paid to the case of a skew-symmetric N-connection ∇N, which means that the torsion of an N-connection considered as a trivalent covariant tensor is skew-symmetric. Such a connection is uniquely defined and corresponds to the endomorphism N = 2ψ, where the endomorphism ψ is defined by the equality ω( X ,Y ) = g (ψX ,Y ) and is called in this work the second structure endomorphism of an almost contact metric manifold. The notion of a ∇N-Einstein almost contact metric manifold is introduced. For the case N = 2ψ, conditions under which almost contact manifolds are ∇N-Einstein manifolds are found.

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9

Mandal, Tarak. "On D-homothetically deformed N(k)-contact metric manifolds." SERIES III - MATEMATICS INFORMATICS PHYSICS 1(63), no. 2 (January 15, 2022): 71–88. http://dx.doi.org/10.31926/but.mif.2021.1.63.2.7.

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In the present paper, we have studied generalized weakly symmetric and generalized weakly Ricci symmetric D-homothetically deformed N(k)-contact metric manifolds. Also we have studied Ricci solitons on deformed N(k)-contact metric manifold and obtained several results if the manifold has generalized weakly symmetric and generalized weakly Ricci symmetric restrictions. We have also proved that there does not exist a Ricci soliton in a D-homothetically deformed N(k )-contact metric manifold. Finally, we give an example.

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10

Erdoğan, Feyza, and Selcen Perktaş. "Lightlike hypersurfaces of an (ε)-para Sasakian manifold with a semi-symmetric non-metric connection." Filomat 32, no. 16 (2018): 5767–86. http://dx.doi.org/10.2298/fil1816767e.

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In the present paper, we study a lightlike hypersurface, when the ambient manifold is an (?)-para Sasakian manifold endowed with a semi-symmetric non-metric connection. We obtain a condition for such a lightlike hypersurface to be totally geodesic. We define invariant and screen semi-invariant lightlike hypersurfaces of (?)-para Sasakian manifolds with a semi-symmetric non-metric connection. Also, we obtain integrability conditions for the distributions D ? ??? and D' ? ??? of a screen semi-invariant lightlike hypersurface of an (?)-para Sasakian manifolds with a semi-symmetric non-metric connection.

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11

SOUZA, FÁBIO S., and PAUL A. SCHWEITZER. "MANIFOLDS THAT ARE NOT LEAVES OF CODIMENSION ONE FOLIATIONS." International Journal of Mathematics 24, no. 14 (December 2013): 1350102. http://dx.doi.org/10.1142/s0129167x13501024.

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We present new open manifolds that are not homeomorphic to leaves of any C0 codimension one foliation of a compact manifold. Among them are simply connected manifolds of dimension d ≥ 5 that are non-periodic in homotopy, namely in their two-dimensional homotopy groups.

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12

FANG, FUQUAN, and XIAOCHUN RONG. "FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS." Communications in Contemporary Mathematics 02, no. 01 (February 2000): 75–86. http://dx.doi.org/10.1142/s0219199700000062.

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We prove a vanishing theorem of certain cohomology classes for an 2n-manifold of finite fundamental group which admits a fixed point free circle action. In particular, it implies that any Tk-action on a compact symplectic manifold of finite fundamental group has a non-empty fixed point set. The vanishing theorem is used to prove two finiteness results in which no lower bound on volume is assumed. (i) The set of symplectic n-manifolds of finite fundamental groups with curvature, λ ≤ sec ≤ Λ, and diameter, diam ; ≤ d, contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. (ii) The set of simply connected n-manifolds (n ≤ 6) with λ ≤ sec ≤ Λ and diam ≤ d contains only finitely many diffeomorphism types depending only on n, λ, Λ and d.

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13

Bukusheva, A. V. "Lifting semi-invariant submanifolds to distribution of almost contact metric manifolds." Differential Geometry of Manifolds of Figures, no. 51 (2020): 39–48. http://dx.doi.org/10.5922/0321-4796-2020-51-5.

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Let M be an almost contact metric manifold of dimension n = 2m + 1. The distribution D of the manifold M admits a natural structure of a smooth manifold of dimension n = 4m + 1. On the manifold M, is defined a linear connection that preserves the distribution D; this connection is determined by the interior connection that allows parallel transport of admissible vectors along admissible curves. The assigment of the linear connection is equivalent to the assignment of a Riemannian metric of the Sasaki type on the distribution D. Certain tensor field of type (1,1) on D defines a so-called prolonged almost contact metric structure. Each section of the distribution D defines a morphism of smooth manifolds. It is proved that if a semi-invariant submanifold of the manifold M and is a covariantly constant vector field with respect to the N-connection , then is a semi-invariant submanifold of the manifold D with respect to the prolonged almost contact metric structure.

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14

ANGELLA, DANIELE, and ADRIANO TOMASSINI. "ON THE COHOMOLOGY OF ALMOST-COMPLEX MANIFOLDS." International Journal of Mathematics 23, no. 02 (February 2012): 1250019. http://dx.doi.org/10.1142/s0129167x11007604.

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Following [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683] and the results obtained by [D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math.36(1) (1976) 225–255] to study the cone of semi-Kähler structures on a compact semi-Kähler manifold.

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15

Fujita, Kento. "Simple normal crossing Fano varieties and log Fano manifolds." Nagoya Mathematical Journal 214 (June 2014): 95–123. http://dx.doi.org/10.1215/00277630-2430136.

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AbstractA projective log variety (X, D) is called a log Fano manifold if X is smooth and if D is a reduced simple normal crossing divisor on Χ with − (KΧ + D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r ≥ n/2 with ρ(X) ≥ 2 or r ≥ n − 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

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16

Fegan, H. D., and B. Steer. "First Order Operators on Manifolds With a Group Action." Canadian Journal of Mathematics 48, no. 4 (August 1, 1996): 758–76. http://dx.doi.org/10.4153/cjm-1996-039-6.

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

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17

ISIDRO, JOSÉ M. "NONCOMMUTATIVE MANIFOLDS FROM THE HIGGS SECTOR OF COINCIDENT D-BRANES." Modern Physics Letters A 20, no. 11 (April 10, 2005): 841–50. http://dx.doi.org/10.1142/s0217732305017123.

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The Higgs sector of the low-energy physics of n coincident D-branes contains the necessary elements for constructing noncommutative manifolds. The coordinates orthogonal to the coincident branes, as well as their conjugate momenta, take values in the Lie algebra of the gauge group living inside the brane stack. In the limit when n→∞ (and in the absence of orientifolds), this is the unitary Lie algebra u(∞). Placing a smooth manifold [Formula: see text] orthogonally to the stack of coincident D-branes, one can construct a noncommutative C⋆-algebra that provides a natural definition of a noncommutative partner for the manifold [Formula: see text].

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18

Blomer, Valentin, and Philippe Michel. "Hybrid bounds for automorphic forms on ellipsoids over number fields." Journal of the Institute of Mathematics of Jussieu 12, no. 4 (December 20, 2012): 727–58. http://dx.doi.org/10.1017/s1474748012000874.

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AbstractWe prove upper bounds for Hecke–Laplace eigenfunctions on certain Riemannian manifolds $X$ of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are $d$-fold products of $2$-spheres or $3$-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a (‘Weyl-type’) saving of $\mathrm{vol} \hspace{0.167em} (X)^{- 1/ 6+ \varepsilon } $.

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19

Chow, Shui-Nee, and Ke ning Lu. "Ck centre unstable manifolds." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 108, no. 3-4 (1988): 303–20. http://dx.doi.org/10.1017/s0308210500014682.

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SynopsisWe consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps. We show that every global centre unstable manifold can be expressed as a graph of a Ck map, provided that the nonlinearities are Ck smooth. The proofs are based on a lemma by D. Henry on a necessary and sufficient condition for a Lipschitz map to be continuously differentiable.

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20

Shukla, S. S., and Akhilesh Yadav. "Radical transversal SCR-lightlike submanifolds of indefinite Sasakian manifolds." Filomat 35, no. 8 (2021): 2585–94. http://dx.doi.org/10.2298/fil2108585s.

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In this paper, we introduce the notion of radical transversal screen Cauchy-Riemann (SCR)- lightlike submanifolds of indefinite Sasakian manifolds giving characterization theorem with some nontrivial examples of such submanifolds. Integrability conditions of distributions D1, D2, D and D? on radical transversal SCR-lightlike submanifolds of an indefinite Sasakian manifold have been obtained. Further, we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic.

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21

Stanciu, Sonia. "D-branes in group manifolds." Journal of High Energy Physics 2000, no. 01 (January 19, 2000): 025. http://dx.doi.org/10.1088/1126-6708/2000/01/025.

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22

Mitsuhiro Kato and Tomoharu Okada. "D-branes on group manifolds." Nuclear Physics B 499, no. 3 (August 1997): 583–95. http://dx.doi.org/10.1016/s0550-3213(97)00305-2.

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23

Angella, Daniele, and Federico Alberto Rossi. "Cohomology of D-complex manifolds." Differential Geometry and its Applications 30, no. 5 (October 2012): 530–47. http://dx.doi.org/10.1016/j.difgeo.2012.07.003.

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24

STOPAR, KRIS. "APPROXIMATION OF HOLOMORPHIC MAPPINGS ON 1-CONVEX DOMAINS." International Journal of Mathematics 24, no. 14 (December 2013): 1350108. http://dx.doi.org/10.1142/s0129167x13501085.

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Let π : Z → X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D ⋐ X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over [Formula: see text] which has prescribed core, it fixes the exceptional set E of D, and is dominating on [Formula: see text]. Each section in this spray is of class [Formula: see text] and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class [Formula: see text] and holomorphic on D can be approximated in the [Formula: see text] topology by π-sections that are holomorphic in open neighborhoods of [Formula: see text]. Under additional assumptions on the submersion we also get approximation by global holomorphic π-sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.

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25

Bukusheva, Aliya V. "Kenmotsu Manifolds with Zero Ricci-Schouten Tensor." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 4 (208) (December 23, 2020): 10–16. http://dx.doi.org/10.18522/1026-2237-2020-4-10-16.

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The paper is dedicated to the investigation of the interior geometry of the Kenmotsu manifolds M. By the interior geometry of the manifold M we mean the aggregate of the properties of the manifold that depend only on the closing of the distribution D of the Kenmotsu manifold as well as on the parallel transport of the vectors from the distribution D along arbitrary curves of the manifold. The invariants of the interior geometry of a Kenmotsu manifold are the following: the Schouten curvature tensor; the 1-form η generating the distribution D; the Lie derivative of the metric tensor g along the structure vector field ; the Schouten-Wagner admissible tensor fields with the components with respect to adapted coordinates; the structural endomorphism φ; the endomorphism N that allows to prolong the interior connection to a connection in a vector bundle. A special attention is payed to the Ricci-Schouten tensor. In particular, it is stated that a Kenmotsu manifold with zero Ricci-Schouten tensor is an Einstein manifold. Conversely, if M is an η-Einstein Kenmotsu manifold and then M is an Einstein manifold with zero Ricci-Schouten tensor. It is proved that the Ricci-Schouten tensor is zero if and only if the Kenmotsu manifold M is locally Ricci-symmetric. This implies the following well-known result: a Kenmotsu manifold is an Einstein manifold if and only if it is locally Ricci-symmetric. An N-connection with torsion, is introduced; this connection is Ricci-flat if and only if M is an Einstein manifold.

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26

Laha, Barnali, Bandana Das, and Arindam Bhattacharyya. "Contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold." Acta Universitatis Sapientiae, Mathematica 5, no. 2 (December 1, 2013): 157–68. http://dx.doi.org/10.2478/ausm-2014-0011.

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Abstract In this paper we prove some properties of the indefinite Lorentzian para-Sasakian manifolds. Section 1 is introductory. In Section 2 we define D-totally geodesic and D⊥-totally geodesic contact CRsubmanifolds of an indefinite Lorentzian para-Sasakian manifold and deduce some results concerning such a manifold. In Section 3 we state and prove some results on mixed totally geodesic contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold. Finally, in Section 4 we obtain a result on the anti-invariant distribution of totally umbilic contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold.

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27

Huang, Jiaxi, and Daniel Tataru. "Local Well-Posedness of Skew Mean Curvature Flow for Small Data in $$d\ge 4$$ Dimensions." Communications in Mathematical Physics 389, no. 3 (January 15, 2022): 1569–645. http://dx.doi.org/10.1007/s00220-021-04303-8.

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AbstractThe skew mean curvature flow is an evolution equation for d dimensional manifolds embedded in $${{\mathbb {R}}}^{d+2}$$ R d + 2 (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $$d\ge 4$$ d ≥ 4 .

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Bonomo, Wescley, and Paulo Varandas. "Continuous flows generate few homeomorphisms." Proceedings of the Edinburgh Mathematical Society 63, no. 4 (August 25, 2020): 971–83. http://dx.doi.org/10.1017/s0013091520000280.

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We describe topological obstructions (involving periodic points, topological entropy and rotation sets) for a homeomorphism on a compact manifold to embed in a continuous flow. We prove that homeomorphisms in a $C^{0}$-open and dense set of homeomorphisms isotopic to the identity in compact manifolds of dimension at least two are not the time-1 map of a continuous flow. Such property is also true for volume-preserving homeomorphisms in compact manifolds of dimension at least five. In the case of conservative homeomorphisms of the torus $\mathbb {T}^{d} (d\ge 2)$ isotopic to identity, we describe necessary conditions for a homeomorphism to be flowable in terms of the rotation sets.

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29

DIMOCK, J. "MARKOV QUANTUM FIELDS ON A MANIFOLD." Reviews in Mathematical Physics 16, no. 02 (March 2004): 243–55. http://dx.doi.org/10.1142/s0129055x04001947.

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We study scalar quantum field theory on a compact manifold. The free theory is defined in terms of functional integrals. For positive mass it is shown to have the Markov property in the sense of Nelson. This property is used to establish a reflection positivity result when the manifold has a reflection symmetry. In dimension d=2 we use the Markov property to establish a sewing operation for manifolds with boundary circles. Also in d=2 the Markov property is proved for interacting fields.

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Planat, Michel, Raymond Aschheim, Marcelo Amaral, and Klee Irwin. "Universal Quantum Computing and Three-Manifolds." Symmetry 10, no. 12 (December 19, 2018): 773. http://dx.doi.org/10.3390/sym10120773.

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A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S 3 . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M 3 . More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group P S L ( 2 , Z ) correspond to d-fold M 3 - coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and M 3 ’s obtained from Dehn fillings are explored.

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Kato, Hisao. "Takens-type reconstruction theorems of one-sided dynamical systems." Nonlinearity 36, no. 3 (February 1, 2023): 1571–92. http://dx.doi.org/10.1088/1361-6544/acb396.

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Abstract The reconstruction theorem deals with dynamical systems that are given by a map T : X → X of a compact metric space X together with an observable f : X → R from X to the real line R . In 1981, by use of Whitney’s embedding theorem, Takens proved that if T : M → M is a (two-sided) diffeomorphism on a compact smooth manifold M with dim M = d , for generic (T, f) there is a bijection between elements x ∈ M and corresponding sequence ( f T j ( x ) ) j = 0 2 d , and moreover, in 2002 Takens proved a generalised version for endomorphisms. In natural sciences and physical engineering, there has been an increase in importance of fractal sets and more complicated spaces, and also in mathematics, many topological and dynamical properties and stochastic analysis of such spaces have been studied. In the present paper, by use of some topological methods we extend the Takens’ reconstruction theorems of compact smooth manifolds to reconstruction theorems of ‘non-invertible’ dynamical systems for a large class of compact metric spaces, which contains PL-manifolds, manifolds with branched structures and some fractal sets, e.g. Menger manifolds, Sierpiński carpet and Sierpiński gasket and dendrites, etc.

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32

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

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

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33

MANEV, MANCHO, and KOSTADIN GRIBACHEV. "A CONNECTION WITH PARALLEL TOTALLY SKEW-SYMMETRIC TORSION ON A CLASS OF ALMOST HYPERCOMPLEX MANIFOLDS WITH HERMITIAN AND ANTI-HERMITIAN METRICS." International Journal of Geometric Methods in Modern Physics 08, no. 01 (February 2011): 115–31. http://dx.doi.org/10.1142/s0219887811005026.

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The subject of investigations are the almost hypercomplex manifolds with Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D and its torsion is totally skew-symmetric. The class of the nearly Kähler manifolds with respect to the first almost complex structure is of special interest. It is proved that D has a D-parallel torsion and is weak if it is not flat. Some curvature properties of these manifolds are studied.

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Barrett, John W., Harald Garcke, and Robert Nürnberg. "Numerical approximation of curve evolutions in Riemannian manifolds." IMA Journal of Numerical Analysis 40, no. 3 (March 7, 2019): 1601–51. http://dx.doi.org/10.1093/imanum/drz012.

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Abstract We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e. conformally equivalent to the Euclidean plane. Examples include the hyperbolic plane, the hyperbolic disc and the elliptic plane, as well as any conformal parameterization of a two-dimensional manifold in ${{\mathbb{R}}}^d$, $d\geqslant 3$. In these spaces we introduce stable numerical schemes for curvature flow and curve diffusion, and we also formulate schemes for elastic flow. Variants of the schemes can also be applied to geometric evolution equations for axisymmetric hypersurfaces in ${{\mathbb{R}}}^d$. Some of the schemes have very good properties with respect to the distribution of mesh points, which is demonstrated with the help of several numerical computations.

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35

Douglas, Michael R., Akishi Kato, and Hirosi Ooguri. "D-brane actions on Kähler manifolds." Advances in Theoretical and Mathematical Physics 1, no. 2 (1997): 237–58. http://dx.doi.org/10.4310/atmp.1997.v1.n2.a3.

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36

Pék, Johanna. "Automorphisms of line element D-manifolds." Acta Mathematica Hungarica 133, no. 1-2 (March 2, 2011): 188–202. http://dx.doi.org/10.1007/s10474-011-0079-1.

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37

Arnaudon, Marc, Anton Thalmaier, and Feng-Yu Wang. "Gradient Estimates on Dirichlet and Neumann Eigenfunctions." International Mathematics Research Notices 2020, no. 20 (September 4, 2018): 7279–305. http://dx.doi.org/10.1093/imrn/rny208.

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Abstract By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c_1(D)$ and $c_2(D)$ for a $d$-dimensional compact Riemannian manifold $D$ with boundary such that $c_1(D)\sqrt \lambda \|\phi \|_\infty \leqslant \|\nabla \phi \|_\infty \leqslant c_2(D)\sqrt \lambda \|\phi \|_\infty $ holds for any Dirichlet eigenfunction $\phi $ of $-\Delta $ with eigenvalue $\lambda $. In particular, when $D$ is convex with nonnegative Ricci curvature, the estimate holds for $c_1(D)= 1/{d\mathrm{e}}$ and $c_2(D)=\sqrt{\mathrm{e}}\left (\frac{\sqrt{2}}{\sqrt{\pi }}+\frac{\sqrt{\pi }}{4\sqrt{2}}\right ).$ Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper.

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38

Grunert, Romain, Wolfgang Kühnel, and Günter Rote. "PL Morse theory in low dimensions." Advances in Geometry 23, no. 1 (January 1, 2023): 135–50. http://dx.doi.org/10.1515/advgeom-2022-0027.

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Abstract We discuss a PL analog of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level; it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong regularity are presented. In particular, we show that in dimensions d ≤ 4 a homologically regular point on a PL d-manifold is always strongly regular. Examples show that this fails in higher dimensions d ≥ 5. One of our constructions involves an embedding of the dunce hat into 4-space and Mazur’s contractible 4-manifold. Finally, decidability questions in this context are discussed.

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39

Fang, Yong, and Patrick Foulon. "On Finsler manifolds of negative flag curvature." Journal of Topology and Analysis 07, no. 03 (May 15, 2015): 483–504. http://dx.doi.org/10.1142/s1793525315500181.

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One of the key differences between Finsler metrics and Riemannian metrics is the non-reversibility, i.e. given two points p and q, the Finsler distance d(p, q) is not necessarily equal to d(q, p). In this paper, we build the main tools to investigate the non-reversibility in the context of large-scale geometry of uniform Finsler Cartan–Hadamard manifolds. In the second part of this paper, we use the large-scale geometry to prove the following dynamical theorem: Let φ be the geodesic flow of a closed negatively curved Finsler manifold. If its Anosov splitting is C2, then its cohomological pressure is equal to its Liouville metric entropy. This result generalizes a previous Riemannian result of U. Hamenstädt.

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40

Calderbank, David M. J. "The Faraday 2-form in Einstein-Weyl geometry." MATHEMATICA SCANDINAVICA 89, no. 1 (September 1, 2001): 97. http://dx.doi.org/10.7146/math.scand.a-14332.

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On a conformal manifold, a compatible torsion free connection $D$ need not be the Levi-Civita connection of a compatible Riemannian metric. The local obstruction is a real $2$-form $F^D$, the Faraday curvature. It is shown that, except in four dimensions, $F^D$ necessarily vanishes if it is divergence free. In four dimensions another differential operator may be applied to $F^D$ to show that an Einstein-Weyl $4$-manifold with selfdual Weyl curvature also has selfdual Faraday curvature and so is either Einstein or locally hypercomplex. More generally, the Bach tensor and the scalar curvature are shown to control the selfduality of $F^D$. Finally, the constancy of the sign of the scalar curvature on compact Einstein-Weyl $4$-manifolds [24] is generalised to higher dimensions. The scalar curvature need not have constant sign in dimensions two and three.

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41

Schmidt, Benjamin, Krishnan Shankar, and Ralf Spatzier. "Almost isotropic Kähler manifolds." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 767 (October 1, 2020): 1–16. http://dx.doi.org/10.1515/crelle-2019-0030.

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AbstractLet M be a complete Riemannian manifold and suppose {p\in M}. For each unit vector {v\in T_{p}M}, the Jacobi operator, {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v}. Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M}. If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds, i.e. manifolds having the property that for each {p\in M}, there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M}. Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}}.

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42

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

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

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43

Ran, Ziv. "A BOGOMOLOV UNOBSTRUCTEDNESS THEOREM FOR LOG-SYMPLECTIC MANIFOLDS IN GENERAL POSITION." Journal of the Institute of Mathematics of Jussieu 19, no. 5 (November 9, 2018): 1509–19. http://dx.doi.org/10.1017/s1474748018000464.

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We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $\unicode[STIX]{x1D6F1}$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(\unicode[STIX]{x1D6F1})$. We prove that $(X,\unicode[STIX]{x1D6F1})$ has unobstructed deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^{2}$ of the open symplectic manifold $X\setminus D(\unicode[STIX]{x1D6F1})$, and in fact coincides with this $H^{2}$ provided the Hodge number $h_{X}^{2,0}=0$, and finally that the degeneracy locus $D(\unicode[STIX]{x1D6F1})$ deforms locally trivially under deformations of $(X,\unicode[STIX]{x1D6F1})$.

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44

HILLMAN, J. A. "-MANIFOLDS." Journal of the Australian Mathematical Society 105, no. 1 (December 4, 2017): 46–56. http://dx.doi.org/10.1017/s1446788717000258.

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We show that closed $\mathbb{S}\text{ol}^{3}\times \mathbb{E}^{1}$-manifolds are Seifert fibred, with general fibre the torus, and base one of the flat 2-orbifolds $T,Kb,\mathbb{A},\mathbb{M}b,S(2,2,2,2),P(2,2)$ or $\mathbb{D}(2,2)$, and outline how such manifolds may be classified.

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45

Özdemir, Nülifer, Şirin Aktay, and Mehmet Solgun. "On Generalized D-Conformal Deformations of Certain Almost Contact Metric Manifolds." Mathematics 7, no. 2 (February 13, 2019): 168. http://dx.doi.org/10.3390/math7020168.

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In this work, we consider almost contact metric manifolds. We investigate the generalized D-conformal deformations of nearly K-cosymplectic, quasi-Sasakian and β -Kenmotsu manifolds. The new Levi–Civita covariant derivative of the new metric corresponding to deformed nearly K-cosymplectic, quasi-Sasakian and β -Kenmotsu manifolds are obtained. Under some restrictions, deformed nearly K-cosymplectic, quasi-Sasakian and β -Kenmotsu manifolds are obtained. Then, the scalar curvature of these three classes of deformed manifolds are analyzed.

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46

Akivis, Maks A., Vladislav V. Goldberg, and Valentin V. Lychagin. "Linearizability of d-webs, d ≥ 4, on two-dimensional manifolds." Selecta Mathematica 10, no. 4 (April 2005): 431–51. http://dx.doi.org/10.1007/s00029-004-0362-x.

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47

Jendoubi, Ch. "On the theory of integral manifolds for some delayed partial differential equations with nondense domain}." Ukrains’kyi Matematychnyi Zhurnal 72, no. 6 (June 17, 2020): 776–89. http://dx.doi.org/10.37863/umzh.v72i6.6020.

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UDC 517.9 Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the nondensely-defined partial differential equation ⅆ u ⅆ t = ( A + B ( t ) ) u ( t ) + f ( t , u t ) , t ∈ R , ( 1 ) where ( A , D ( A ) ) satisfies the Hille – Yosida condition, ( B ( t ) ) t ∈ R is a family of operators in ℒ ( D ( A ) ¯ , X ) satisfying some measurability and boundedness conditions, and the nonlinear forcing term f satisfies ‖ f ( t , ϕ ) - f ( t , ψ ) ‖ ≤ φ ( t ) ‖ ϕ - ψ ‖ 𝒞 , here, φ belongs to some admissible spaces and ϕ , ψ ∈ 𝒞 : = C ( [ - r ,0 ] , X ) . We first present an exponential convergence result between the stable manifold and every mild solution of (1). Then we prove the existence of center-unstable manifolds for such solutions.Our main methods are invoked by the extrapolation theory and the Lyapunov – Perron method based on the admissible functions properties.

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48

McReynolds, D. B. "Geometric Spectra and Commensurability." Canadian Journal of Mathematics 67, no. 1 (January 2013): 184–97. http://dx.doi.org/10.4153/cjm-2014-003-9.

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AbstractThe work of Reid, Chinburg–Hamilton–Long–Reid, Prasad–Rapinchuk, and the author with Reid have demonstrated that geodesics or totally geodesic submanifolds can sometimes be used to determine the commensurability class of an arithmetic manifold. The main results of this article show that generalizations of these results to other arithmetic manifolds will require a wide range of data. Specifically, we prove that certain incommensurable arithmetic manifolds arising from the semisimple Lie groups of the form (SL(d, R)) r(SL(d, C))s have the same commensurability classes of totally geodesic submanifolds coming from a fixed field. This construction is algebraic and shows the failure of determining, in general, a central simple algebra from subalgebras over a fixed field. This, in turn, can be viewed in terms of forms of SLd and the failure of determining the form via certain classes of algebraic subgroups.

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49

Taleshian, A., A. A. Hosseinzadeh, and F. Khaniani. "On D-conformal Curvature Tensor Sasakian Manifolds." Journal of Nonlinear Analysis and Application 2011 (2011): 1–5. http://dx.doi.org/10.5899/2011/jnaa-00088.

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50

De, Krishnendu. "D-Homothetic deformation of LP-Sasakian manifolds." Novi Sad Journal of Mathematics 45, no. 2 (December 20, 2015): 113–23. http://dx.doi.org/10.30755/nsjom.2014.033.

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