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Journal articles on the topic 'Submanifolds theory'

Author: Grafiati

Published: 14 March 2023

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1

Jain, Varun, Amrinder Pal Singh, and Rakesh Kumar. "On the geometry of lightlike submanifolds of indefinite statistical manifolds." International Journal of Geometric Methods in Modern Physics 17, no. 07 (June 2020): 2050099. http://dx.doi.org/10.1142/s0219887820500991.

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We study lightlike submanifolds of indefinite statistical manifolds. Contrary to the classical theory of submanifolds of statistical manifolds, lightlike submanifolds of indefinite statistical manifolds need not to be statistical submanifold. Therefore, we obtain some conditions for a lightlike submanifold of indefinite statistical manifolds to be a lightlike statistical submanifold. We derive the expression of statistical sectional curvature and finally obtain some conditions for the induced statistical Ricci tensor on a lightlike submanifold of indefinite statistical manifolds to be symmetric.

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2

YANG, GUO-HONG, SHI-XIANG FENG, GUANG-JIONG NI, and YI-SHI DUAN. "RELATIONS OF TWO TRANSVERSAL SUBMANIFOLDS AND GLOBAL MANIFOLD." International Journal of Modern Physics A 16, no. 21 (August 20, 2001): 3535–51. http://dx.doi.org/10.1142/s0217751x01005080.

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In Riemann geometry, the relations of two transversal submanifolds and global manifold are discussed without any concrete models. By replacing the normal vector of a submanifold with the tangent vector of another submanifold, the metric tensors, Christoffel symbols and curvature tensors of the three manifolds are connected at the intersection points of the two submanifolds. When the inner product of the two tangent vectors of submanifolds vanishes, some corollaries of these relations give the most important second fundamental form and Gauss–Codazzi equation in the conventional submanifold theory. As a special case, the global manifold which is Euclidean is considered. It is pointed out that, in order to obtain the nonzero energy–momentum tensor of matter field in a submanifold, there must be the contributions of the above inner product and the other submanifold. Generally speaking, a submanifold is closely related to the matter fields of the other submanifold and the two submanifolds affect each other through the above inner product.

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3

MONTE, EDMUNDO M. "MATHEMATICAL SUPPORT TO BRANEWORLD THEORY." International Journal of Geometric Methods in Modern Physics 04, no. 08 (December 2007): 1259–67. http://dx.doi.org/10.1142/s0219887807002557.

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The braneworld theory appear with the purpose of solving the problem of the hierarchy of the fundamental interactions. The perspectives of the theory emerge as a new physics, for example, deviation of the law of Newton's gravity. One of the principles of the theory is to suppose that the braneworld is local submanifold in a space of high dimension, the bulk, solution of Einstein's equations in high dimension. In this paper we approach the mathematical consistency of this theory with a new proof of the fundamental theorem of submanifolds for the case of semi-Riemannian manifolds. This theorem consists of an essential mathematical support for this new theory. We find the integrability conditions for the existence of space–time submanifolds in a pseudo-Euclidean space.

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4

Wu, B. Y. "Some results on Finsler submanifolds." International Journal of Mathematics 27, no. 03 (March 2016): 1650021. http://dx.doi.org/10.1142/s0129167x1650021x.

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In this paper we study the submanifold theory in terms of Chern connection. We introduce the notions of the second fundamental form and mean curvature for Finsler submanifolds, and establish the fundamental equations by means of moving frame for the hypersurface case. We provide the estimation of image radius for compact submanifold, and prove that there exists no compact minimal submanifold in any complete noncompact and simply connected Finsler manifold with nonpositive flag curvature. We also characterize the Minkowski hyperplanes, Minkowski hyperspheres and Minkowski cylinders as the only hypersurfaces in Minkowski space with parallel second fundamental form.

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5

Varolin, Dror. "A Takayama-type extension theorem." Compositio Mathematica 144, no. 2 (March 2008): 522–40. http://dx.doi.org/10.1112/s0010437x07002989.

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AbstractWe prove a theorem on the extension of holomorphic sections of powers of adjoint bundles from submanifolds of complex codimension 1 having non-trivial normal bundle. The first such result, due to Takayama, considers the case where the canonical bundle is twisted by a line bundle that is a sum of a big and nef line bundle and a $\mathbb {Q}$-divisor that has Kawamata log terminal singularities on the submanifold from which extension occurs. In this paper we weaken the positivity assumptions on the twisting line bundle to what we believe to be the minimal positivity hypotheses. The main new idea is an L2 extension theorem of Ohsawa–Takegoshi type, in which twisted canonical sections are extended from submanifolds with non-trivial normal bundle.

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6

Ali, Akram, Wan Othman, and Sayyadah Qasem. "Geometric inequalities for CR-warped product submanifolds of locally conformal almost cosymplectic manifolds." Filomat 33, no. 3 (2019): 741–48. http://dx.doi.org/10.2298/fil1903741a.

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In this paper, we establish some inequalities for the squared norm of the second fundamental form and the warping function of warped product submanifolds in locally conformal almost cosymplectic manifolds with pointwise ?-sectional curvature. The equality cases are also considered. Moreover, we prove a triviality result for CR-warped product submanifold by using the integration theory on a compact orientate manifold without boundary.

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7

Biard, Séverine, and Emil J. Straube. "L2-Sobolev theory for the complex Green operator." International Journal of Mathematics 28, no. 09 (August 2017): 1740006. http://dx.doi.org/10.1142/s0129167x17400067.

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These notes are concerned with the [Formula: see text]-Sobolev theory of the complex Green operator on pseudoconvex, oriented, bounded and closed Cauchy–Riemann (CR)-submanifolds of [Formula: see text] of hypersurface type. This class of submanifolds generalizes that of boundaries of pseudoconvex domains. We first discuss briefly the CR-geometry of general CR-submanifolds and then specialize to this class. Next, we review the basic [Formula: see text]-theory of the tangential CR operator and the associated complex Green operator(s) on these submanifolds. After these preparations, we discuss recent results on compactness and regularity in Sobolev spaces of the complex Green operator(s).

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8

Urban, Zbyněk, and Ján Brajerčík. "The fundamental Lepage form in variational theory for submanifolds." International Journal of Geometric Methods in Modern Physics 15, no. 06 (May 8, 2018): 1850103. http://dx.doi.org/10.1142/s0219887818501037.

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The multiple-integral variational functionals for finite-dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert form in the classical mechanics. The notion of a Lepage form is extended to manifolds of regular velocities and plays a basic role in formulation of the variational theory for submanifolds. The theory is illustrated on the minimal submanifolds problem, including analysis of conservation law equations.

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9

Mihai, Ion, and Radu-Ioan Mihai. "A New Algebraic Inequality and Some Applications in Submanifold Theory." Mathematics 9, no. 11 (May 23, 2021): 1175. http://dx.doi.org/10.3390/math9111175.

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We give a simple proof of the Chen inequality involving the Chen invariant δ(k) of submanifolds in Riemannian space forms. We derive Chen’s first inequality and the Chen–Ricci inequality. Additionally, we establish a corresponding inequality for statistical submanifolds.

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10

Urban, Zbyněk, and Demeter Krupka. "Foundations of higher-order variational theory on Grassmann fibrations." International Journal of Geometric Methods in Modern Physics 11, no. 07 (August 2014): 1460023. http://dx.doi.org/10.1142/s0219887814600238.

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A setting for higher-order global variational analysis on Grassmann fibrations is presented. The integral variational principles for one-dimensional immersed submanifolds are introduced by means of differential 1-forms with specific properties, similar to the Lepage forms from the variational calculus on fibred manifolds. Prolongations of immersions and vector fields to the Grassmann fibrations are defined as a geometric tool for the variations of immersions, and the first variation formula in the infinitesimal form is derived. Its consequences, the Euler–Lagrange equations for submanifolds and the Noether theorem on invariant variational functionals are proved. Examples clarifying the meaning of the Noether theorem in the context of variational principles for submanifolds are given.

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11

Albers, Peter, and Doris Hein. "Cuplength estimates in Morse cohomology." Journal of Topology and Analysis 08, no. 02 (March 15, 2016): 243–72. http://dx.doi.org/10.1142/s1793525316500102.

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The main goal of this paper is to give a unified treatment to many known cuplength estimates with a view towards Floer theory. As the base case, we prove that for [Formula: see text]-perturbations of a function which is Morse–Bott along a closed submanifold, the number of critical points is bounded below in terms of the cuplength of that critical submanifold. As we work with rather general assumptions the proof also applies in a variety of Floer settings. For example, this proves lower bounds (some of which were known) for the number of fixed points of Hamiltonian diffeomorphisms, Hamiltonian chords for Lagrangian submanifolds, translated points of contactomorphisms, and solutions to a Dirac-type equation.

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12

Li, Yanlin, Mohan Khatri, Jay Prakash Singh, and Sudhakar K. Chaubey. "Improved Chen’s Inequalities for Submanifolds of Generalized Sasakian-Space-Forms." Axioms 11, no. 7 (July 1, 2022): 324. http://dx.doi.org/10.3390/axioms11070324.

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In this article, we derive Chen’s inequalities involving Chen’s δ-invariant δM, Riemannian invariant δ(m1,⋯,mk), Ricci curvature, Riemannian invariant Θk(2≤k≤m), the scalar curvature and the squared of the mean curvature for submanifolds of generalized Sasakian-space-forms endowed with a quarter-symmetric connection. As an application of the obtain inequality, we first derived the Chen inequality for the bi-slant submanifold of generalized Sasakian-space-forms.

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13

Izumiya, Shyuichi, and Masaki Kasedou. "Lightlike flat geometry of spacelike submanifolds in Lorentz–Minkowski space." International Journal of Geometric Methods in Modern Physics 11, no. 05 (May 2014): 1450049. http://dx.doi.org/10.1142/s0219887814500492.

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In this paper, we investigate differential geometry on spacelike submanifolds in Lorentz–Minkowski space from the viewpoint of contact with lightlike hyperplanes. It is called the lightlike flat geometry which has been well established for the codimension-two case. In order to develop the theory for the general codimension-case, we introduce the notion of codimension-two spacelike canal submanifolds which is a main tool in this paper. We apply the theory of Lagrangian/Legendrian singularities to codimension-two spacelike canal submanifolds and obtain the relation with the previous results on the codimension-two case.

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14

DAJCZER, MARCOS, and RUY TOJEIRO. "AN EXTENSION OF THE CLASSICAL RIBAUCOUR TRANSFORMATION." Proceedings of the London Mathematical Society 85, no. 1 (March 2002): 211–32. http://dx.doi.org/10.1112/s0024611502013552.

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We extend the notion of Ribaucour transformation from classical surface theory to the theory of holonomic submanifolds of pseudo-Riemannian space forms with arbitrary dimension and codimension, that is, submanifolds with flat normal bundle admitting a global system of principal coordinates. Bianchi gave a positive answer to the question of whether among the Ribaucour transforms of a surface with constant mean or Gaussian curvature there exist other surfaces with the same property. Our main achievement is to solve the same problem for the class of holonomic submanifolds with constant sectional curvature. 2000 Mathematical Subject Classification: 53B25, 58J72.

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15

BEJANCU, Aurel, and Hani Reda FARRAN. "THEORY OF FINSLER SUBMANIFOLDS VIA BERWALD CONNECTION." International Electronic Journal of Geometry 7, no. 1 (April 30, 2014): 108–25. http://dx.doi.org/10.36890/iejg.594500.

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16

Wang, Zhigang, Donghe Pei, and Liang Chen. "Geometry of 1-lightlike submanifolds in anti-de Sitter n-space." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 143, no. 5 (September 25, 2013): 1089–113. http://dx.doi.org/10.1017/s0308210512000558.

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In this paper we investigate the differential geometry of 1-lightlike submanifolds in anti-de Sitter n-space as an application of the theory of Legendrian singularities. Based on some theory of lightlike submanifolds, we also introduce the notion of 1-lightlike horospherical Gauss curvature, which is important for us to study the singularities of 1-lightlike horospherical hypersurfaces. Moreover, we discuss the related geometric property of 1-lightlike horospherical hypersurfaces in anti-de Sitter n-space.

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17

Kauffman, Louis H., and Eiji Ogasa. "Brieskorn submanifolds, local moves on knots, and knot products." Journal of Knot Theory and Its Ramifications 28, no. 10 (September 2019): 1950068. http://dx.doi.org/10.1142/s0218216519500688.

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We first prove the following: Let [Formula: see text] and [Formula: see text]. Let [Formula: see text] and [Formula: see text] be closed, oriented, [Formula: see text]-dimensional [Formula: see text]-connected, simple submanifolds of [Formula: see text]. Then [Formula: see text] and [Formula: see text] are isotopic if and only if a Seifert matrix associated with a simple Seifert hypersurface for [Formula: see text] is [Formula: see text]-[Formula: see text]-equivalent to that for [Formula: see text]. We also discuss the [Formula: see text] case. This result implies one of our main results: Let [Formula: see text]. A 1-link [Formula: see text] is pass-equivalent to a 1-link [Formula: see text] if and only if [Formula: see text] is [Formula: see text]-pass-equivalent to [Formula: see text]. Here, [Formula: see text] means the knot product of [Formula: see text] and [Formula: see text], and [Formula: see text] means [Formula: see text]. See the body of the paper for the definition of knot products. It also implies the other main results: We strengthen the authors’ old result that two-fold cyclic suspension commutes with the performance of the twist move for spherical [Formula: see text]-knots. See the body for the precise statement. Furthermore, it implies the following: Let [Formula: see text] and [Formula: see text]. Let [Formula: see text] be a closed oriented [Formula: see text]-submanifold of [Formula: see text]. Then [Formula: see text] is a Brieskorn submanifold if and only if [Formula: see text] is [Formula: see text]-connected, simple and has a [Formula: see text]-Seifert matrix associated with a simple Seifert hypersurface that is [Formula: see text]-[Formula: see text]-equivalent to a [Formula: see text]-type (see the body of the paper for a definition). We also discuss the [Formula: see text] case.

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18

Chrastinová, Veronika, and Václav Tryhuk. "Automorphisms of Submanifolds." Advances in Difference Equations 2010, no. 1 (2010): 202731. http://dx.doi.org/10.1155/2010/202731.

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19

Kashani, S. M. B. "Isoparametric functions and submanifolds." Glasgow Mathematical Journal 35, no. 2 (May 1993): 145–52. http://dx.doi.org/10.1017/s0017089500009691.

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The theory of isoparametric functions and a family of isoparametric hypersurfaces began essentially with E. Cartan in 1930's. He defined a real valued function V defined on a Riemannian space form to be isoparametric if ∥grad υ∥2=TV and ΔV = SV for some real valued functions S, T. Then a family of hypersurfaces Mt, is called isoparametric if Mt,=V-1 (t) where t is a regular value of V.

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20

FIORI, SIMONE, and ROBERTO ROSSI. "STIEFEL-MANIFOLD LEARNING BY IMPROVED RIGID-BODY THEORY APPLIED TO ICA." International Journal of Neural Systems 13, no. 05 (October 2003): 273–90. http://dx.doi.org/10.1142/s0129065703001625.

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In previous contributions we presented a new class of algorithms for orthonormal learning of a linear neural network with p inputs and m outputs, based on the equations describing the dynamics of a massive rigid frame in a submanifold of ℛp. While exhibiting interesting features, such as intrinsic numerical stability, strongly binding to the orthonormal submanifolds, and good controllability of the learning dynamics, tested on principal/independent component analysis, the proposed algorithms were not completely satisfactory from a computational-complexity point of view. The main drawback was the need to repeatedly evaluate a matrix exponential map. With the aim to lessen the computational efforts pertaining to these algorithms, we propose here an improvement based on the closed-form Rodriguez formula for the exponential map. Such formula is available in the p=3 and m=3 case, which is discussed with details here. In particular, experimental results on independent component analysis (ICA), carried out with both synthetic and real-world data, help confirming the computational gain due to the proposed improvement.

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21

FU, HAIPING, and HONGWEI XU. "VANISHING AND TOPOLOGICAL SPHERE THEOREMS FOR SUBMANIFOLDS IN A HYPERBOLIC SPACE." International Journal of Mathematics 19, no. 07 (August 2008): 811–22. http://dx.doi.org/10.1142/s0129167x0800490x.

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We extend the vanishing and sphere theorems due to Lawson, Simons, Xin, Shiohama and Xu. By using the techniques of calculus of variations in the geometric measure theory, we prove the vanishing theorem for homology groups of submanifolds in the hyperbolic space Hn(c) with negative constant curvature c. Moreover, we obtain a topological sphere theorem for certain complete submanifolds in Hn(c).

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22

Zambon, Marco. "An example of coisotropic submanifolds C1-close to a given coisotropic submanifold." Differential Geometry and its Applications 26, no. 6 (December 2008): 635–37. http://dx.doi.org/10.1016/j.difgeo.2008.04.011.

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23

Goto, Shin-itiro. "Nonequilibrium thermodynamic process with hysteresis and metastable states—A contact Hamiltonian with unstable and stable segments of a Legendre submanifold." Journal of Mathematical Physics 63, no. 5 (May 1, 2022): 053302. http://dx.doi.org/10.1063/5.0062495.

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In this paper, a dynamical process in a statistical thermodynamic system of spins exhibiting a phase transition is described on a contact manifold, where such a dynamical process is a process that a metastable equilibrium state evolves into the most stable symmetry broken equilibrium state. Metastable and the most stable equilibrium states in the symmetry broken phase or ordered phase are assumed to be described as pruned projections of Legendre submanifolds of contact manifolds, where these pruned projections of the submanifolds express hysteresis and pseudo-free energy curves. Singularities associated with phase transitions are naturally arose in this framework as has been suggested by Legendre singularity theory. Then, a particular contact Hamiltonian vector field is proposed so that a pruned segment of the projected Legendre submanifold is a stable fixed point set in a region of a contact manifold and that another pruned segment is a unstable fixed point set. This contact Hamiltonian vector field is identified with a dynamical process departing from a metastable equilibrium state to the most stable equilibrium one. To show the statements above explicitly, an Ising type spin model with long-range interactions, called the Husimi–Temperley model, is focused, where this model exhibits a phase transition.

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24

Müller, Stefan. "C0-characterization of symplectic and contact embeddings and Lagrangian rigidity." International Journal of Mathematics 30, no. 09 (August 2019): 1950035. http://dx.doi.org/10.1142/s0129167x19500356.

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We present a novel [Formula: see text]-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of [Formula: see text]-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from [Formula: see text]-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a [Formula: see text]-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of [Formula: see text]-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

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25

Jin, Dae Ho, and Jae Won Lee. "Generic Lightlike Submanifolds of an Indefinite Cosymplectic Manifold." Mathematical Problems in Engineering 2011 (2011): 1–16. http://dx.doi.org/10.1155/2011/610986.

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Lightlike geometry has its applications in general relativity, particularly in black hole theory. Indeed, it is known that lightlike hypersurfaces are examples of physical models of Killing horizons in general relativity (Galloway, 2007). In this paper, we introduce the definition of generic lightlike submanifolds of an indefinite cosymplectic manifold. We investigate new results on a class of generic lightlike submanifolds of an indefinite cosymplectic manifold .

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26

Monte, E. M. "The Fundamental theorem of Submanifolds in Braneworld Theory." EAS Publications Series 30 (2008): 377–81. http://dx.doi.org/10.1051/eas:0830065.

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27

Bernig, Andreas, Dmitry Faifman, and Gil Solanes. "Crofton formulas in pseudo-Riemannian space forms." Compositio Mathematica 158, no. 10 (October 2022): 1935–79. http://dx.doi.org/10.1112/s0010437x22007722.

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Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz–Killing curvature integrals of a submanifold with corners, to be computed by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

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28

Chen, Bang-Yen, Adara M. Blaga, and Gabriel-Eduard Vîlcu. "Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants." Mathematics 10, no. 4 (February 14, 2022): 591. http://dx.doi.org/10.3390/math10040591.

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One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of δ-invariants for Riemannian manifolds, which are different in nature from the classical curvature invariants. The earlier results on δ-invariants and their applications have been summarized in the first author’s book published in 2011 Pseudo-Riemannian Geometry, δ-Invariants and Applications (ISBN: 978-981-4329-63-7). In this survey, we present a comprehensive account of the development of the differential geometry of submanifolds in complex space forms involving the δ-invariants done mostly after the publication of the book.

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Hadfield, Charles, Santosh Kandel, and Michele Schiavina. "Ruelle Zeta Function from Field Theory." Annales Henri Poincaré 21, no. 12 (October 6, 2020): 3835–67. http://dx.doi.org/10.1007/s00023-020-00964-8.

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Abstract We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function for BF theory when an unusual gauge-fixing condition on contact manifolds is imposed. This suggests an alternative rephrasing of a conjecture due to Fried on the equivalence between Ruelle zeta function and analytic torsion, in terms of homotopies of Lagrangian submanifolds.

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Polterovich, Leonid. "Symplectic displacement energy for Lagrangian submanifolds." Ergodic Theory and Dynamical Systems 13, no. 2 (June 1993): 357–67. http://dx.doi.org/10.1017/s0143385700007410.

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AbstractRecently H. Hofer defined a new symplectic invariant which has a beautiful dynamical meaning. In the present paper we study this invariant for Lagrangian submanifolds of symplectic manifolds. Our approach is based on Gromov's theory of pseudo-holomorphic curves.

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31

de Carvalho, A., S. Chion, and M. Dajczer. "Conformal Kaehler Euclidean submanifolds." Differential Geometry and its Applications 82 (June 2022): 101893. http://dx.doi.org/10.1016/j.difgeo.2022.101893.

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32

VIRO, OLEG. "TWISTED ACYCLICITY OF A CIRCLE AND SIGNATURES OF A LINK." Journal of Knot Theory and Its Ramifications 18, no. 06 (June 2009): 729–55. http://dx.doi.org/10.1142/s0218216509007142.

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Homology of the circle with non-trivial local coefficients is trivial. From this well-known fact we deduce geometric corollaries involving codimension-two links. In particular, the Murasugi–Tristram signatures are extended to invariants of links formed of arbitrary oriented closed codimension two submanifolds of an odd-dimensional sphere. The novelty is that the submanifolds are not assumed to be disjoint, but are transversal to each other, and the signatures are parametrized by points of the whole torus. Murasugi–Tristram inequalities and their generalizations are also extended to this setup.

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Djuretic, Jovana, Jelena Katic, and Darko Milinkovic. "Comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory." Filomat 30, no. 5 (2016): 1161–74. http://dx.doi.org/10.2298/fil1605161d.

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We compare spectral invariants in periodic orbits and Lagrangian Floer homology case, for a closed symplectic manifold P and its closed Lagrangian submanifolds L, when ?|?2(P,L)=0, and ?|?2(P,L)=0. We define a product HF*(H)?HF*(H:L) ? HF*(H:L) and prove subadditivity of invariants with respect to this product.

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34

Tasaki, Hiroyuki. "Integral geometry under cut loci in compact symmetric spaces." Nagoya Mathematical Journal 137 (March 1995): 33–53. http://dx.doi.org/10.1017/s0027763000005067.

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The theory of integral geometry has mainly treated identities between integral invariants of submanifolds in Riemannian homogeneous spaces like as dμg(g) where M and N are submanifolds in a Riemannian homogeneous spaces of a Lie group G and I(M ∩ gN) is an integral invariant of M ∩ gN. For example Poincaré’s formula is one of typical identities in integral geometry, which is as follows. We denote by M(R2) the identity component of the group of isometries of the plane R2 with a suitable invariant measure μM(R2).

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Singh, Jai Pratap, and Kripa Sindhu Prasad. "ON SUBMANIFOLDS OF A MANIFOLD ADMITTING fa(2ν + 3, −1) - STRUCTURE." South East Asian J. of Mathematics and Mathematical Sciences 18, no. 03 (December 30, 2022): 171–78. http://dx.doi.org/10.56827/seajmms.2022.1803.14.

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Psomopoulou defined and studied the Invariant submanifolds of a man- ifold with f (2ν + 3, 1)-structure. In this paper fa(2ν + 3, 1) structure has been defined and submanifolds, of a manifold with such a structure have been studied. Some interesting results have been stated and proved in this paper.

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Dajczer, Marcos, and Miguel Ibieta Jimenez. "Conformal infinitesimal variations of submanifolds." Differential Geometry and its Applications 75 (April 2021): 101721. http://dx.doi.org/10.1016/j.difgeo.2021.101721.

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37

Wang, Sung Ho. "Calibrated lifts of minimal submanifolds." Differential Geometry and its Applications 23, no. 3 (November 2005): 351–60. http://dx.doi.org/10.1016/j.difgeo.2005.06.007.

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38

Borisenko, A., and V. Rovenski. "About topology of saddle submanifolds." Differential Geometry and its Applications 25, no. 2 (April 2007): 220–33. http://dx.doi.org/10.1016/j.difgeo.2006.08.001.

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39

Mirzoyan, V. A. "Structure theorems for Ricci-semisymmetric submanifolds and geometric description of a class of minimal semi-Einstein submanifolds." Sbornik: Mathematics 197, no. 7 (August 31, 2006): 997–1024. http://dx.doi.org/10.1070/sm2006v197n07abeh003786.

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Hou, Zhong Hua, Wei Shi, and Muhammad Hanif. "Submanifolds in SE(3) and their applications in envelope theory." Mechanism and Machine Theory 142 (December 2019): 103572. http://dx.doi.org/10.1016/j.mechmachtheory.2019.07.012.

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41

AMORIM, LINO, YONG–GEUN OH, and JOANA OLIVEIRA DOS SANTOS. "Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry–Mather theory." Mathematical Proceedings of the Cambridge Philosophical Society 165, no. 3 (August 31, 2017): 411–34. http://dx.doi.org/10.1017/s0305004117000561.

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AbstractWe construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants, developed by Oh, and results, by Abouzaid, about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalisation of graph selector. We then, following Bernard–Oliveira dos Santos, use these results to give a new characterisation of the Aubry and Mañé sets of a Tonelli Hamiltonian and to generalise a result of Arnaud on Lagrangians invariant under the flow of such Hamiltonians.

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Bagaglini, Leonardo. "Non-orientable three-submanifolds of G2-manifolds." Advances in Geometry 19, no. 3 (July 26, 2019): 401–14. http://dx.doi.org/10.1515/advgeom-2018-0023.

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Abstract By analogy with associative and co-associative cases, we study a class of three-dimensional non-orientable submanifolds of manifolds with a G2-structure, modelled on planes lying in aspecial G2-orbit. An application of the Cartan–Kähler theory shows that some three-manifolds can be presented in this way. We also classify all the homogeneous ones in ℝℙ7.

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Neilson, Peter D., Megan D. Neilson, and Robin T. Bye. "A Riemannian Geometry Theory of Synergy Selection for Visually-Guided Movement." Vision 5, no. 2 (May 25, 2021): 26. http://dx.doi.org/10.3390/vision5020026.

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Bringing together a Riemannian geometry account of visual space with a complementary account of human movement synergies we present a neurally-feasible computational formulation of visuomotor task performance. This cohesive geometric theory addresses inherent nonlinear complications underlying the match between a visual goal and an optimal action to achieve that goal: (i) the warped geometry of visual space causes the position, size, outline, curvature, velocity and acceleration of images to change with changes in the place and orientation of the head, (ii) the relationship between head place and body posture is ill-defined, and (iii) mass-inertia loads on muscles vary with body configuration and affect the planning of minimum-effort movement. We describe a partitioned visuospatial memory consisting of the warped posture-and-place-encoded images of the environment, including images of visible body parts. We depict synergies as low-dimensional submanifolds embedded in the warped posture-and-place manifold of the body. A task-appropriate synergy corresponds to a submanifold containing those postures and places that match the posture-and-place-encoded visual images that encompass the required visual goal. We set out a reinforcement learning process that tunes an error-reducing association memory network to minimize any mismatch, thereby coupling visual goals with compatible movement synergies. A simulation of a two-degrees-of-freedom arm illustrates that, despite warping of both visual space and posture space, there exists a smooth one-to-one and onto invertible mapping between vision and proprioception.

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Zhang, Xue-shan. "Nonexistence of stable currents in submanifolds of a product of two spheres." Bulletin of the Australian Mathematical Society 44, no. 2 (October 1991): 325–36. http://dx.doi.org/10.1017/s0004972700029762.

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By using techniques of the calculus of variations in geometric measure theory, we investigate the nonexistence of stable integral currents in Sn1 × Sn2 and its immersed submanifolds. Several vanishing theorems concerning the homology group of these manifolds are established.

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Mirzoyan, V. A. "Submanifolds with semiparallel tensor fields as envelopes." Sbornik: Mathematics 193, no. 10 (October 31, 2002): 1493–505. http://dx.doi.org/10.1070/sm2002v193n10abeh000686.

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46

FISH, JOEL W. "ESTIMATES FOR J-CURVES AS SUBMANIFOLDS." International Journal of Mathematics 22, no. 10 (October 2011): 1375–431. http://dx.doi.org/10.1142/s0129167x11007306.

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In this paper, we develop some basic analytic tools to study compactness properties of J-curves (i.e. pseudoholomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves by establishing an extrinsic monotonicity principle for nonnegative functions f satisfying Δf ≥ -c2f, we show that curves locally parametrized as a graph over a coordinate tangent plane have all derivatives a priori bounded in terms of curvature and ambient geometry, and we thus establish ϵ-regularity for the square length of their second fundamental forms. These results are all provided for J-curves either with or without Lagrangian boundary and hold in almost all Hermitian manifolds of arbitrary even dimension (i.e. Riemannian manifolds for which the almost complex structure is an isometry).

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Heinzner, Peter, Gerald W. Schwarz, and Henrik Stötzel. "Stratifications with respect to actions of real reductive groups." Compositio Mathematica 144, no. 1 (January 2008): 163–85. http://dx.doi.org/10.1112/s0010437x07003259.

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AbstractWe study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group $G^{\mathbb {C}}$ and that with respect to a compatible maximal compact subgroup U of $G^{\mathbb {C}}$ the action on Z is Hamiltonian. There is a corresponding gradient map $\mu _{\mathfrak {p}}\colon X\to \mathfrak {p}^*$ where $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ is a Cartan decomposition of $\mathfrak {g}$. We obtain a Morse-like function $\eta _{\mathfrak {p}}:=\Vert \mu _{\mathfrak {p}}\Vert ^2$ on X. Associated with critical points of $\eta _{\mathfrak {p}}$ are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds Sβ of X which are called pre-strata. In cases where $\mu _{\mathfrak {p}}$ is proper, the pre-strata form a decomposition of X and in cases where X is compact they are the strata of a Morse-type stratification of X. Our results are generalizations of results of Kirwan obtained in the case where $G=U^{\mathbb {C}}$ and X=Z is compact.

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YANG, GUO-HONG. "THE TRANSVERSAL SUBMANIFOLDS AND GAUSS–BONNET–CHERN THEOREM." Modern Physics Letters A 13, no. 26 (August 30, 1998): 2123–30. http://dx.doi.org/10.1142/s0217732398002242.

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From general ϕ-mapping method and transverse theory, the Gauss–Bonnet–Chern theorem is discussed in detail. Through the calculation of the generalized Winding number, i.e. the degree of Gauss map, the Euler–Poincaré characteristic is decomposed in terms of two transversal submanifolds. This decomposition is a further detailed study of Gauss–Bonnet–Chern theorem and Poincaré–Hopf index theorem.

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49

Ranicki, Andrew. "The L-Theory of Twisted Quadratic Extensions." Canadian Journal of Mathematics 39, no. 2 (April 1, 1987): 345–64. http://dx.doi.org/10.4153/cjm-1987-017-x.

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For surgery on codimension 1 submanifolds with non-trivial normal bundle the theory of Wall [13, Section 12C] has obstruction groups LN∗(π′ → π), with π a group and π′ a subgroup of index 2, such that there is defined an exact sequence involving the ordinary L-groups of rings with involutionwith the superscript w signifying a different involution on Z[π]. Geometry was used in [13] to identifywith (α, u) an antistructure on Z[π′] in the sense of Wall [14]. The main result of this paper is a purely algebraic version of this identification, for any twisted quadratic extension of a ring with antistructure.

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Seyyedali, Reza, and Gábor Székelyhidi. "Extremal metrics on blowups along submanifolds." Journal of Differential Geometry 114, no. 1 (January 2020): 171–92. http://dx.doi.org/10.4310/jdg/1577502024.

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