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Journal articles on the topic 'Degree of a Submanifold'

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Published: 10 March 2023

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1

BANGERT, VICTOR, and XIAOJUN CUI. "Calibrations and laminations." Mathematical Proceedings of the Cambridge Philosophical Society 162, no. 1 (June 15, 2016): 151–71. http://dx.doi.org/10.1017/s0305004116000475.

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AbstractA calibration of degree k ∈ ℕ on a Riemannian manifold M is a closed differential k-form θ such that the integral of θ over every k-dimensional, oriented submanifold N is smaller or equal to the Riemannian volume of N. A calibration θ is said to calibrate N if θ restricts to the oriented volume form of N. We investigate conditions on a calibration θ that ensure the existence of submanifolds calibrated by θ. The cases k = 1 and k > 1 turn out to be essentially different. Our main result says that, on a compact manifold M, a calibration θ calibrates a lamination if θ is simple, of class C1, and if θ has minimal comass norm in its cohomology class.

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2

Little, Robert D. "Regular cyclic actions on complex projective space with codimension-two fixed points." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 65, no. 1 (August 1998): 51–67. http://dx.doi.org/10.1017/s1446788700039392.

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AbstractIfM2nis a cohomologyCPnandPis an odd prime, letGpbe the cyclic group of orderp. A TypeI I0Gpaction onM2nis an action with fixed point set a codimension-2 submanifold and an isolated point. A TypeI I0Gpaction is standard if it is regular and the degree of the fixed codimension-2 submanifold is one. If n is odd and M2nadmits a standardGpaction of TypeI I0, then every TypeI I0GpactionM2nis standard and so, if n is odd,CPnadmits aGpaction of TypeI I0if and only if the action is standard.

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3

LEMAIRE, L., and J. C. WOOD. "ON THE SPACE OF HARMONIC 2-SPHERES IN CP2." International Journal of Mathematics 07, no. 02 (April 1996): 211–25. http://dx.doi.org/10.1142/s0129167x96000128.

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Carrying further the work of T.A. Crawford, we show that each component of the space of harmonic maps from the 2-sphere to complex projective 2-space of degree d and energy 4πE is a smooth closed submanifold of the space of all Cj maps (j≥2). We achieve this by showing that the Gauss transform which relates them to spaces of holomorphic maps of given degree and ramification index is smooth and has injective differential.

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4

Wen, Wen-Yu. "Thermodynamic metric of deformed Schwarzschild black holes." International Journal of Modern Physics D 26, no. 10 (August 20, 2017): 1750106. http://dx.doi.org/10.1142/s0218271817501061.

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Thermodynamic metric usually works only for those black holes with more than one conserved charge, thereby excluding the Schwarzschild black hole. In this paper, however, different versions of thermodynamic metric are computed and compared for the Schwarzschild-like black hole by introducing new degrees of freedom. These new degrees of freedom have two purposes. First, the deformed metric may be treated offshell to the ordinary Schwarzschild black hole, and onshell physics corresponds to the submanifold by gauge fixing of this additional degree of freedom. In particular, the thermal Ricci scalar for the Schwarzschild black hole, though different for various deformations, can be obtained by switching off the deformation. Second, while deformed metric is treated onshell, a divergent Ricci scalar may signal an exotic phase in which new physical degrees of freedom manifest. This paper considers the new degree of freedom as the running Newton constant, a cutoff scale for regular black holes, a noncommutative deformation or the deformed parameter in the nonextensive Tsallis–Rènyi entropy.

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van der Schaft, Arjan, and Bernhard Maschke. "Geometry of Thermodynamic Processes." Entropy 20, no. 12 (December 4, 2018): 925. http://dx.doi.org/10.3390/e20120925.

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Since the 1970s, contact geometry has been recognized as an appropriate framework for the geometric formulation of thermodynamic systems, and in particular their state properties. More recently it has been shown how the symplectization of contact manifolds provides a new vantage point; enabling, among other things, to switch easily between the energy and entropy representations of a thermodynamic system. In the present paper, this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally-defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, is extended to the definition of port-thermodynamic systems and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control.

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Liao, Qizheng, and J. Michael McCarthy. "On the Seven Position Synthesis of a 5-SS Platform Linkage." Journal of Mechanical Design 123, no. 1 (November 1, 1997): 74–79. http://dx.doi.org/10.1115/1.1330269.

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This paper builds on Innocenti’s polynomial solution for the 5-SS platform that generates a one-degree of freedom movement through seven specified spatial positions of a rigid body. We show that his 60×60 resultant can be reduced to one that is 10×10. We then actuate the linkage using a prismatic joint on the sixth leg and determine the trajectory of the reference point through the specified positions. The singularity submanifold of this associated 6-SS platform provides information about the movement characteristics of the 5-SS linkage.

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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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8

Zhang, Yang, Bo Pang, Xiankai Li, and Gang Chen. "Aerodynamic Shape Optimization with Grassmannian Shape Parameterization Method." Energies 15, no. 20 (October 19, 2022): 7722. http://dx.doi.org/10.3390/en15207722.

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The conventional method of optimizing the aerodynamic performance of an airfoil heavily depends on the confines of the design space. The design variables create a non-normalized space that is fragmented into several different clusters of airfoils. An approach that is data-driven and deforms airfoils over a Grassmannian submanifold is utilized in the work that is being presented here. The affine deformation, which includes camber and thickness, can be uncoupled from the method that is currently in use, and the operations that are performed on the airfoil shape can be made smooth enough to prevent unreasonable shapes from being produced. The CST method is also a part of the current study so that a comparison can be made between the two. A new method to describe the airfoil geometries over the Grassmannian space was generated using a dataset that contained 7007 different shapes of airfoils. These two methods are used to parameterize the subsonic (NACA0012) and transonic (RAE2822) airfoils, and the new method cuts the number of design variables from twelve to six, resulting in a reduction in overall complexity. The findings demonstrate that the new method maintains a high degree of consistency regardless of the flow conditions.

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Lárusson, Finnur. "Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds." Canadian Journal of Mathematics 52, no. 5 (October 1, 2000): 982–98. http://dx.doi.org/10.4153/cjm-2000-041-0.

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AbstractLet Y be an infinite covering space of a projective manifold M in N of dimension n ≥ 2. Let C be the intersection with M of at most n − 1 generic hypersurfaces of degree d in N. The preimage X of C in Y is a connected submanifold. Let φ be the smoothed distance from a fixed point in Y in a metric pulled up from M. Let φ(X) be the Hilbert space of holomorphic functions f on X such that f2e−φ is integrable on X, and define φ(Y) similarly. Our main result is that (under more general hypotheses than described here) the restriction φ(Y) → φ(X) is an isomorphism for d large enough.This yields new examples of Riemann surfaces and domains of holomorphy in n with corona. We consider the important special case when Y is the unit ball in n, and show that for d large enough, every bounded holomorphic function on X extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on . Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from X to .

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10

Du, Li. "Classification of f-biharmonic submanifolds in Lorentz space forms." Open Mathematics 19, no. 1 (January 1, 2021): 1299–314. http://dx.doi.org/10.1515/math-2021-0084.

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Abstract In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

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Qiu, Qiang, and Qixin Cao. "Task constrained motion planning for 7-degree of freedom manipulators with parameterized submanifolds." Industrial Robot: An International Journal 45, no. 3 (May 21, 2018): 363–70. http://dx.doi.org/10.1108/ir-01-2018-0004.

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PurposeThis paper aims to use the redundancy of a 7-DOF (degree of freedom) serial manipulator to solve motion planning problems along a given 6D Cartesian tool path, in the presence of geometric constraints, namely, obstacles and joint limits.Design/methodology/approachThis paper describes an explicit expression of the task submanifolds for a 7-DOF redundant robot, and the submanifolds can be parameterized by two parameters with this explicit expression. Therefore, the global search method can find the feasible path on this parameterized graph.FindingsThe proposed planning algorithm is resolution complete and resolution optimal for 7-DOF manipulators, and the planned path can satisfy task constraint as well as avoiding singularity and collision. The experiments on Motoman SDA robot are reported to show the effectiveness.Research limitations/implicationsThis algorithm is still time-consuming, and it can be improved by applying parallel collision detection method or lazy collision detection, adopting new constraints and implementing more effective graph search algorithms.Originality/valueCompared with other task constrained planning methods, the proposed algorithm archives better performance. This method finds the explicit expression of the two-dimensional task sub-manifolds, so it’s resolution complete and resolution optimal.

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12

Wang, Shi. "On splitting rank of non-compact-type symmetric spaces and bounded cohomology." Journal of Topology and Analysis 12, no. 02 (September 5, 2018): 465–89. http://dx.doi.org/10.1142/s179352531950050x.

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Let [Formula: see text] be a higher rank symmetric space of non-compact type where [Formula: see text]. We define the splitting rank of [Formula: see text], denoted by [Formula: see text], to be the maximal dimension of a totally geodesic submanifold [Formula: see text] which splits off an isometric [Formula: see text]-factor. We compute explicitly the splitting rank for each irreducible symmetric space. For an arbitrary (not necessarily irreducible) symmetric space, we show that the comparison map [Formula: see text] is surjective in degrees [Formula: see text], provided [Formula: see text] has no direct factors of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. This generalizes the result of [J.-F. Lafont and S. Wang, Barycentric straightening and bounded cohomology, to appear in J. Eur. Math. Soc.] regarding Dupont’s problem.

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13

Yamamura, M., and A. Kuriyama. "Equations of Collective Submanifold for Large Amplitude Collective Motion and Its Coupling with Intrinsic Degrees of Freedom. III." Progress of Theoretical Physics 77, no. 1 (January 1, 1987): 94–105. http://dx.doi.org/10.1143/ptp.77.94.

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14

Yamamura, M., and A. Kuriyama. "Equations of Collective Submanifold for Large Amplitude Collective Motion and Its Coupling with Intrinsic Degrees of Freedom. I." Progress of Theoretical Physics 75, no. 2 (February 1, 1986): 272–87. http://dx.doi.org/10.1143/ptp.75.272.

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15

Yamamura, M., and A. Kuriyama. "Equations of Collective Submanifold for Large Amplitude Collective Motion and Its Coupling with Intrinsic Degrees of Freedom. II." Progress of Theoretical Physics 75, no. 3 (March 1, 1986): 583–91. http://dx.doi.org/10.1143/ptp.75.583.

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16

Dovermann, Karl, and Robert D. Little. "Cohomology complex projective space with degree one codimension-two fixed submanifolds." Pacific Journal of Mathematics 173, no. 2 (April 1, 1996): 387–401. http://dx.doi.org/10.2140/pjm.1996.173.387.

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17

Iuras¸cu, C. C., and F. C. Park. "Geometric Algorithms for Kinematic Calibration of Robots Containing Closed Loops." Journal of Mechanical Design 125, no. 1 (March 1, 2003): 23–32. http://dx.doi.org/10.1115/1.1539512.

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We present a coordinate-invariant, differential geometric formulation of the kinematic calibration problem for a general class of mechanisms. The mechanisms considered may have multiple closed loops, be redundantly actuated, and include an arbitrary number of passive joints that may or may not be equipped with joint encoders. Some form of measurement information on the position and orientation of the tool frame may also be available. Our approach rests on viewing the joint configuration space of the mechanism as an embedded submanifold of an ambient manifold, and formulating error measures in terms of the Riemannian metric specified in the ambient manifold. Based on this geometric framework, we pose the kinematic calibration problem as one of determining a parametrized multidimensional surface that is a best fit (in the sense of the chosen metric) to a given set of measured points in both the ambient and task space manifolds. Several optimization algorithms that address the various possibilities with respect to available measurement data and choice of error measures are given. Experimental and simulation results are given for the Eclipse, a six degree-of-freedom redundantly actuated parallel mechanism. The geometric framework and algorithms presented in this article have the desirable feature of being invariant with respect to the local coordinate representation of the forward and inverse kinematics and of the loop closure equations, and also provide a high-level framework in which to classify existing approaches to kinematic calibration.

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Giovannardi, Gianmarco. "Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds." Analysis and Geometry in Metric Spaces 8, no. 1 (July 1, 2020): 68–91. http://dx.doi.org/10.1515/agms-2020-0105.

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AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

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Okonek, Christian, and Andrei Teleman. "A wall-crossing formula for degrees of Real central projections." International Journal of Mathematics 25, no. 04 (April 2014): 1450038. http://dx.doi.org/10.1142/s0129167x14500384.

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The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

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Arimoto, Suguru, and Masahiro Sekimoto. "Natural Resolution of Ill-Posed Inverse Kinematics for Redundant Robots: A Challenge to Bernstein’s Degrees-of-Freedom Problem." Journal of Robotics and Mechatronics 18, no. 5 (October 20, 2006): 651–60. http://dx.doi.org/10.20965/jrm.2006.p0651.

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Over half a century ago, A. N. Bernstein observed that “dexterity” in human limb movement emerges from the involvement of multijoint motion with surplus degrees of freedom (DOF). Robotics posits that DOF redundancy in robot may enhance dexterity and versatility. Kinematic redundancy involves the problem of ill-posed inverse kinematics from task-description space to joint space. This problem is conventionally avoided by introducing an artificial performance index and uniquely determining an inverse kinematics solution by minimizing it. Instead of taking this conventional avoidance solution, we propose challenging Bernstein’s DOF problem by introducing two direct novel concepts - stability on a manifold and transferability to a submanifold - in dealing with human multijoint movement in reaching and showing that sensory feedback from task space to joint space together with adequate damping (joint velocity feedback) enables any solution to overall closed-loop dynamics to converge naturally and coordinately to a lower-dimensional manifold describing a set of joint states fulfilling a given motion task. This means that a reaching task is accomplished by sensory feedback with the appropriate choice of a stiffness parameter and damping coefficients without having to consider inverse kinematics. We also show that these concepts cope with the annoying “variability” of redundant joint motion seen typically in skilled human reaching. In conclusion, we propose a virtual spring/damper hypothesis that leads to natural control of skilled movement in redundant multijoint reaching.

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21

Melhem, Khoder, Zhaoheng Liu, and Antonio Loría. "A Unified Framework for Dynamics and Lyapunov Stability of Holonomically Constrained Rigid Bodies." Journal of Advanced Computational Intelligence and Intelligent Informatics 9, no. 4 (July 20, 2005): 387–94. http://dx.doi.org/10.20965/jaciii.2005.p0387.

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A new dynamic model for interconnected rigid bodies is proposed here. The model formulation makes it possible to treat any physical system with finite number of degrees of freedom in a unified framework. This new model is a nonminimal realization of the system dynamics since it contains more state variables than is needed. A useful discussion shows how the dimension of the state of this model can be reduced by eliminating the redundancy in the equations of motion, thus obtaining the minimal realization of the system dynamics. With this formulation, we can for the first time explicitly determine the equations of the constraints between the elements of the mechanical system corresponding to the interconnected rigid bodies in question. One of the advantages coming with this model is that we can use it to demonstrate that Lyapunov stability and control structure for the constrained system can be deducted by projection in the submanifold of movement from appropriate Lyapunov stability and stabilizing control of the corresponding unconstrained system. This procedure is tested by some simulations using the model of two-link planar robot.

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22

Wang, Lizhou. "Singular critical points for variational free boundary problem from isoparametric hypersurfaces." Communications in Contemporary Mathematics 18, no. 03 (March 22, 2016): 1650010. http://dx.doi.org/10.1142/s0219199716500103.

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We construct three families of singular critical points for a variational free boundary problem. These critical points are homogeneous solutions of degree one to some overdetermined boundary value problem. The intersections of the level sets of these solutions with the unit sphere are isoparametric hypersurfaces and their focal submanifolds.

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23

Szalai, Robert, David Ehrhardt, and George Haller. "Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2202 (June 2017): 20160759. http://dx.doi.org/10.1098/rspa.2016.0759.

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In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude–frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here, a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. This model identification utilizes Taken’s delay-embedding theorem, as well as a least square fit to the Taylor expansion of the sampling map associated with that embedding. The SSMs are then constructed for the sampling map using the parametrization method for invariant manifolds, which assumes that the manifold is an embedding of, rather than a graph over, a spectral subspace. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy.

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24

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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Domingo-Juan, M. Carmen, and Vicente Miquel. "Bounding the First Dirichlet Eigenvalue of a Tube Around a Complex Submanifold of ℂP n (λ) in Terms of the Degrees of the Polynomials Defining It." Journal of Geometric Analysis 24, no. 1 (July 18, 2012): 92–103. http://dx.doi.org/10.1007/s12220-012-9328-y.

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Ya Su, Sheng Li, Shengjin Wang, and Yun Fu. "Submanifold Decomposition." IEEE Transactions on Circuits and Systems for Video Technology 24, no. 11 (November 2014): 1885–97. http://dx.doi.org/10.1109/tcsvt.2014.2329375.

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Larsen, Jens C. "Submanifold geometry." Journal of Geometric Analysis 4, no. 2 (June 1994): 179–205. http://dx.doi.org/10.1007/bf02921546.

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Siddiqi, M. D., A. Haseeb, and M. Ahmad. "Skew semi-invariant submanifolds of generalized quasi-Sasakian manifolds." Carpathian Mathematical Publications 9, no. 2 (January 2, 2018): 188–97. http://dx.doi.org/10.15330/cmp.9.2.188-197.

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In the present paper, we study a new class of submanifolds of a generalized Quasi-Sasakian manifold, called skew semi-invariant submanifold. We obtain integrability conditions of the distributions on a skew semi-invariant submanifold and also find the condition for a skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold to be mixed totally geodesic. Also it is shown that a skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold will be anti-invariant if and only if $A_{\xi}=0$; and the submanifold will be skew semi-invariant submanifold if $\nabla w=0$. The equivalence relations for the skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold are given. Furthermore, we have proved that a skew semi-invariant $\xi^\perp$-submanifold of a normal almost contact metric manifold and a generalized Quasi-Sasakian manifold with non-trivial invariant distribution is $CR$-manifold. An example of dimension 5 is given to show that a skew semi-invariant $\xi^\perp$ submanifold is a $CR$-structure on the manifold.

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Gelbukh, Irina. "Realization of a Graph as the Reeb Graph of a Morse–Bott or a Round Function." Studia Scientiarum Mathematicarum Hungarica 59, no. 1 (April 1, 2022): 1–16. http://dx.doi.org/10.1556/012.2022.01512.

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We prove criteria for a graph to be the Reeb graph of a function of a given class on a closed manifold: Morse–Bott, round, and in general smooth functions whose critical set consists of a finite number of submanifolds. The criteria are given in terms of whether the graph admits an orientation, which we call S-good orientation, with certain conditions on the degree of sources and sinks, similar to the known notion of good orientation in the context of Morse functions. We also study when such a function is the height function associated with an immersion of the manifold. The condition for a graph to admit an S-good orientation can be expressed in terms of the leaf blocks of the graph.

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Mao, Jing, and Shaodong Qin. "On pinching theorems for compact pseudo-umbilical submanifold." Demonstratio Mathematica 45, no. 3 (September 1, 2012): 645–54. http://dx.doi.org/10.1515/dema-2013-0390.

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AbstractConsider submanifolds in the nested space. For a compact pseudoumbilical submanifold with parallel mean curvature vector of a Riemannian submanifold with constant curvature immersed in a quasi-constant curvature Riemannian manifold, two sufficient conditions are given to let the pseudo-umbilical submanifold become a totally umbilical submanifold.

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Dirik, Suleyman, Mehmet Atceken, and Umit Yildirim. "PSEUDO-SLANT SUBMANIFOLD IN KENMOTSU SPACE FORMS." JOURNAL OF ADVANCES IN MATHEMATICS 11, no. 10 (January 16, 2016): 5680–96. http://dx.doi.org/10.24297/jam.v11i10.774.

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In this paper, we study geometry of the pseudo-slant submanifold of a Kenmotsu space form. Necessary and suffcient conditions are given for a submanifold to be a pseudo-slant submanifold in Kenmotsu manifolds. Finally, we give some results for totally umbilical pseudo-slant submanifold in a Kenmotsu manifold and Kenmotsu space form.

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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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Olver, Peter J. "Invariant submanifold flows." Journal of Physics A: Mathematical and Theoretical 41, no. 34 (August 11, 2008): 344017. http://dx.doi.org/10.1088/1751-8113/41/34/344017.

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Xu, Senlin, and Qinglan Xia. "Conformai deformation of a close Riemannian submanifold to minimal submanifold." Chinese Science Bulletin 43, no. 6 (March 1998): 527. http://dx.doi.org/10.1007/bf02883827.

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HEINTZE, E., C. OLMOS, and G. THORBERGSSON. "SUBMANIFOLDS WITH CONSTANT PRINCIPAL CURVATURES AND NORMAL HOLONOMY GROUPS." International Journal of Mathematics 02, no. 02 (April 1991): 167–75. http://dx.doi.org/10.1142/s0129167x91000107.

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A submanifold has by definition constant principal curvatures if the eigenvalues of the shape operators Aξ are constant for any parallel normal field ξ along any curve. It is shown that a submanifold of Euclidean space has constant principal curvatures if and only if it is an isoparametric or a focal manifold of an isoparametric submanifold. Furthermore a "homogeneous slice theorem" is proved which says that the fibres of the projection of an isoparametric submanifold onto a full focal manifold are homogeneous isoparametric. To this end it is shown that any two points of an isoparametric submanifold can be connected by a piecewise differentiable curve whose pieces are tangent to one of the simultaneous eigenspaces Ei, i ∈ I, of the shape operators provided that the corresponding reflections generate the Weyl group of the isoparametric submanifold.

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36

Bashir, Mohammed Ali. "Chen's problem on mixed foliate CR-submanifolds." Bulletin of the Australian Mathematical Society 40, no. 1 (August 1989): 157–60. http://dx.doi.org/10.1017/s0004972700003609.

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We prove that the simply connected compact mixed foliate CR-submanifold in a hyperbolic complex space form is either a complex submanifold or a totally real submanifold. This is the problem posed by Chen.

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37

Khan, Kamran, and Viqar Khan. "Some characterizing results for hemi-slant warped product submanifolds of a Kaehler manifold." Filomat 33, no. 11 (2019): 3615–25. http://dx.doi.org/10.2298/fil1911615k.

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Many differential geometric properties of submanifolds of a Kaehler manifold are looked into via canonical structure tensors P and F on the submanifold. For instance, a CR-submanifold of a Kaehler manifold is a CR-product (i.e. locally a Riemannian product of a holomorphic and a totally real submanifold) if and only if the canonical tensor P is parallel on the submanifold. Since, warped product manifolds are generalized version of Riemannian product of manifolds, in this article, we consider the covariant derivatives of the structure tensors on a hemi-slant submanifold of a Kaehler manifold. Our investigations have led us to characterize hemi-slant warped product submanifolds.

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38

Dursunf, Uǧur. "On product k-Chen submanifolds." Glasgow Mathematical Journal 39, no. 3 (September 1997): 243–49. http://dx.doi.org/10.1017/s0017089500032171.

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B. Rouxel [7] and S. J. Li and C. S. Houh [6] have generalised the notion of an -submanifold (Chen submanifold) to an k-submanifold. In [1] we have studied the relation between their definitions for the Euclidean case.

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39

Devgan, Anu, and R. K. Nagaich. "Totally Contact Umbilical Radical Transversal Lightlike Submanifolds Of An Almost Contact Manifold With B-Metric." JOURNAL OF ADVANCES IN MATHEMATICS 13, no. 4 (July 3, 2017): 7286–94. http://dx.doi.org/10.24297/jam.v13i4.6234.

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In the present paper, we study the geometry of totally contact umbilical radical transversal lightlike submanifolds and totally contact umbilical CR- submanifold of an indenite Sasaki-like almost contact manifold with B-metric. We nd the necessary and sucient condition for the characterization of the induced connection to be a metric connection. Finally, we have proved that for a totally contact umbilical CR-submanifold, totally contact umbilical radical transversal lightlike submanifold is a totally geodesic radical transversal lightlike submanifold.

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40

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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41

Ishan, Amira A. "Submanifolds of Sasakian Manifolds with Certain Parallel Operators." ISRN Geometry 2014 (February 2, 2014): 1–4. http://dx.doi.org/10.1155/2014/821869.

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We study submanifolds of Sasakian manifolds and obtain a condition under which certain naturally defined symmetric tensor field on the submanifold is to be parallel and use this result to obtain conditions under which a submanifold of the Sasakian manifold is an invariant submanifold.

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42

Chen, Bang-Yen. "A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds." Axioms 8, no. 4 (October 30, 2019): 120. http://dx.doi.org/10.3390/axioms8040120.

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A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden–Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper, we provide a comprehensive survey on this important class of submanifolds.

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Şahin, Fulya. "On the topology of CR-warped product submanifolds." International Journal of Geometric Methods in Modern Physics 15, no. 02 (January 24, 2018): 1850032. http://dx.doi.org/10.1142/s0219887818500329.

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We obtain a necessary condition for homology group to be zero on CR-warped product submanifold in Euclidean spaces in terms of second fundamental form of the submanifold and warping function. By using this condition, we show that such CR-warped product submanifold is a homotopy sphere.

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Arslan, Kadri, Bengü Bayram, Betül Bulca, and Günay Öztürk. "Rotational submanifolds in Euclidean spaces." International Journal of Geometric Methods in Modern Physics 16, no. 02 (February 2019): 1950029. http://dx.doi.org/10.1142/s0219887819500294.

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The rotational embedded submanifold was first studied by Kuiper as a submanifold in [Formula: see text]. The generalized Beltrami submanifolds and toroidal submanifold are the special examples of these kind of submanifolds. In this paper, we consider [Formula: see text]-dimensional rotational embedded submanifolds in Euclidean [Formula: see text]-space [Formula: see text]. We give some basic curvature properties of this type of submanifolds. Further, we obtain some results related with the scalar curvature and mean curvature of these submanifolds. As an application, we give an example of rotational submanifold in [Formula: see text].

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VAISMAN, IZU. "TANGENT DIRAC STRUCTURES AND SUBMANIFOLDS." International Journal of Geometric Methods in Modern Physics 02, no. 05 (October 2005): 759–75. http://dx.doi.org/10.1142/s0219887805000843.

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We write down the local equations that characterize the submanifolds N of a Dirac manifold M which have a normal bundle that is either a coisotropic or an isotropic submanifold of TM endowed with the tangent Dirac structure. In the Poisson case, these formulas once again prove a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of TM with the tangent Poisson structure if and only if N is a Dirac submanifold. In the presymplectic case it is the isotropy of the normal bundle which characterizes the corresponding notion of a Dirac submanifold. On the way, we give a simple definition of the tangent Dirac structure, make new remarks about it and establish its characteristic, local formulas for various interesting classes of submanifolds of a Dirac manifold.

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Gupta, Ram Shankar. "Screen slant lightlike submanifolds of indefinite cosymplectic manifolds." gmj 18, no. 1 (March 2011): 83–97. http://dx.doi.org/10.1515/gmj.2011.0012.

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Abstract In this paper, we introduce the notion of a screen slant lightlike submanifold of an indefinite cosymplectic manifold. We provide a characterization theorem for the existence of a screen slant lightlike submanifold with examples. Also, we give an example of a minimal screen slant lightlike submanifold of and prove some characterization theorems.

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47

Atc̣eken, Mehmet. "Semi-Slant Submanifolds of an Almost Paracontact Metric Manifold." Canadian Mathematical Bulletin 53, no. 2 (June 1, 2010): 206–17. http://dx.doi.org/10.4153/cmb-2010-003-0.

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AbstractIn this paper, we define and study the geometry of semi-slant submanifolds of an almost paracontact metric manifold. We give some characterizations for a submanifold to be semi-slant submanifold to be semi-slant product and obtain integrability conditions for the distributions involved in the definition of a semi-slant submanifold.

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Gök, Mustafa, and Erol Kılıç. "Non-invariant submanifolds of locally decomposable golden Riemannian manifolds." Arabian Journal of Mathematics 10, no. 1 (January 12, 2021): 77–89. http://dx.doi.org/10.1007/s40065-020-00307-9.

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AbstractIn this paper, we investigate any non-invariant submanifold of a locally decomposable golden Riemannian manifold in the case that the rank of the set of tangent vector fields of the induced structure on the submanifold by the golden structure of the ambient manifold is less than or equal to the codimension of the submanifold.

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Gök, Mustafa, Sadık Keleş, and Erol Kılıç. "Some Characterizations of Semi-Invariant Submanifolds of Golden Riemannian Manifolds." Mathematics 7, no. 12 (December 9, 2019): 1209. http://dx.doi.org/10.3390/math7121209.

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In this paper, we study some characterizations for any submanifold of a golden Riemannian manifold to be semi-invariant in terms of canonical structures on the submanifold, induced by the golden structure of the ambient manifold. Besides, we determine forms of the distributions involved in the characterizations of a semi-invariant submanifold on both its tangent and normal bundles.

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HAIDER, S. M. KHURSHEED, V. A. KHAN, and S. I. HUSAIN. "TOTALLY UMBILICAL CR-SUBMANIFOLDS OF A KAEHLER MANIFOLD." Tamkang Journal of Mathematics 24, no. 1 (March 1, 1993): 43–49. http://dx.doi.org/10.5556/j.tkjm.24.1993.4473.

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In the present paper we study totally umbilical CR-submanifolds of a Kaehler manifold. A classification theorem for a $D^\perp$-totally umbilical CR-submanifold is proved. The conditions under which a CR- submanifold becomes a CR-product are obtained, and finally a theorem for a CR-submanifold to be a proper CR-product is also established.

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