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Journal articles on the topic 'Flat submanifold'

Author: Grafiati

Published: 28 May 2022

Last updated: 31 July 2025

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1

Blair, David E. "On Lagrangian Catenoids." Canadian Mathematical Bulletin 50, no. 3 (2007): 321–33. http://dx.doi.org/10.4153/cmb-2007-031-4.

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AbstractRecently I. Castro and F.Urbano introduced the Lagrangian catenoid. Topologically, it is ℝ × Sn–1 and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ℂn is foliated by round (n – 1)-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ℂn. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to t

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Kumar, Sangeet, Rakesh Kumar, and R. K. Nagaich. "Characterization of Holomorphic Bisectional Curvature ofGCR-Lightlike Submanifolds." Advances in Mathematical Physics 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/356263.

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We obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of aGCR-lightlike submanifold of an indefinite Kaehler manifold. We discuss the boundedness of holomorphic sectional curvature ofGCR-lightlike submanifolds of an indefinite complex space form. We establish a condition for aGCR-lightlike submanifold of an indefinite complex space form to be null holomorphically flat. We also obtain some characterization theorems for holomorphic sectional and holomorphic bisectional curvature.

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Ali A. Shihab and Mihail Banaru. "ON SPECTRA OF SOME TENSORS OF SIX-DIMENSINAL KÄHLERIAN AND NEARLY-KÄHLERIAN SUBMANIFOLDS OF CAYLEY ALGEBRA." Tikrit Journal of Pure Science 20, no. 4 (2023): 42–147. http://dx.doi.org/10.25130/tjps.v20i4.1227.

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Six-dimensional Kählerian and nearly-Kählerian submanifolds of Cayley algebra are considered. Spectra of some classical tensors of such submanifolds of the octave algebra are computed. It is proved that a nearly-Kählerian six-dimensional submanifold of Cayley algebra is conharmonically flat if and only if it is holomorphically isometric to the complex Euclidean space with a canonical Kählerian structure.

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4

Bashir, M. A. "On Totally Real Submanifolds in a 6-Sphere." Canadian Mathematical Bulletin 33, no. 2 (1990): 162–66. http://dx.doi.org/10.4153/cmb-1990-027-4.

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AbstractThe 6-dimensional sphere S6 has an almost complex structure induced by properties of Cayley algebra. With respect to this structure S6 is a nearly Kaehlerian manifold. We investigate 2-dimensional totally real submanifolds in S6. We prove that a 2-dimensional totally real submanifold in S6 is flat.

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Kim, Jihun, Jeonghyeong Park, and Bayram Şahin. "Weakly Einstein Equivalence in a Golden Space Form and Certain CR-submanifolds." Mathematica Slovaca 73, no. 6 (2023): 1615–27. http://dx.doi.org/10.1515/ms-2023-0117.

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ABSTRACT It is known that the Golden space form may not be an Einstein manifold. In this paper, it is shown that the condition to be Einstein for a Golden space form is equivalent to being weakly Einstein. In addition, the partial geodesic and cyclic parallelism of the CR-submanifolds of a Golden space are examined, and the case of the constant Golden sectional curvature is determined. Moreover, the CR-submanifolds with semi-flat normal connection are studied and an inequality is obtained. The equality case of this inequality is also checked. We also consider the totally umbilical CR-submanifo

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Aquib, Mohd, Michel Nguiffo Boyom, Mohammad Hasan Shahid, and Gabriel-Eduard Vîlcu. "The First Fundamental Equation and Generalized Wintgen-Type Inequalities for Submanifolds in Generalized Space Forms." Mathematics 7, no. 12 (2019): 1151. http://dx.doi.org/10.3390/math7121151.

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In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some applications in various slant cases. Finally, we obtain obstructions to the existence of non-flat generalized complex space forms and non-flat generalized Sasakian space forms in terms of dimension of the vector space of solutions to the first fundamental equation on such spaces.

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Upreti, Jaya, and Shankar Lal. "On H-Projectively flat KH-Structure submanifold." Journal of the Tensor Society 1, no. 01 (2007): 41–49. http://dx.doi.org/10.56424/jts.v1i01.9947.

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In this paper we have obtained the condition for a submanifold of a KH-structure manifold to be KH-structure. A few relations are found in the second fundamental tensor H and k. Using these results we have also find out the flatness of H-projective curvature tensor in KH-structure submanifold.

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8

Shugailo, Olena. "On curve based ruled affine submanifolds." Proceedings of the International Geometry Center 17, no. 3 (2024): 203–17. https://doi.org/10.15673/pigc.v17i3.2883.

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In this paper we consider affine ruled submanifolds of arbitrary dimension and codimension in the classical sense, i.e. curve based ones. For such a submanifold we define the natural parameterization and the natural transversal distribution. We calculate all the affine characteristics for such an affine immersion. We find conditions on the base curve and the directions of the rectilinear generators such that the induced connection is flat and the natural transversal distribution is equiaffine.

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9

Ghazal, Tahsin, and Sharief Deshmukh. "Submanifolds of Euclidean space with parallel mean curvature vector." International Journal of Mathematics and Mathematical Sciences 14, no. 3 (1991): 533–36. http://dx.doi.org/10.1155/s0161171291000728.

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The object of the paper is to study some compact submanifolds in the Euclidean spaceRnwhose mean curvature vector is parallel in the normal bundle. First we prove that there does not exist ann-dimensional compact simply connected totally real submanifold inR2nwhose mean curvature vector is parallel. Then we show that then-dimensional compact totally real submanifolds of constant curvature and parallel mean curvature inR2nare flat. Finally we show that compact Positively curved submanifolds inRnwith parallel mean curvature vector are homology spheres. The last result in particular for even dime

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10

Mihai, Ion, and Yoshihiko Tazawa. "On 3-dimensional contact slant submanifolds in Sasakian space forms." Bulletin of the Australian Mathematical Society 68, no. 2 (2003): 275–83. http://dx.doi.org/10.1017/s0004972700037655.

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Recently, B.-Y. Chen obtained an inequality for slant surfaces in complex space forms. Further, B.-Y. Chen and one of the present authors proved the non-minimality of proper slant surfaces in non-flat complex space forms. In the present paper, we investigate 3-dimensional proper contact slant submanifolds in Sasakian space forms. A sharp inequality is obtained between the scalar curvature (intrinsic invariant) and the main extrinsic invariant, namely the squared mean curvature.It is also shown that a 3-dimensional contact slant submanifold M of a Sasakian space form M̆(c), with c ≠ 1, cannot b

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11

KON, MAYUKO. "PINCHING THEOREMS FOR A COMPACT MINIMAL SUBMANIFOLD IN A COMPLEX PROJECTIVE SPACE." Bulletin of the Australian Mathematical Society 77, no. 1 (2008): 99–114. http://dx.doi.org/10.1017/s0004972708000129.

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AbstractWe give a formula for the Laplacian of the second fundamental form of an n-dimensional compact minimal submanifold M in a complex projective space CPm. As an application of this formula, we prove that M is a geodesic minimal hypersphere in CPm if the sectional curvature satisfies K≥1/n, if the normal connection is flat, and if M satisfies an additional condition which is automatically satisfied when M is a CR submanifold. We also prove that M is the complex projective space CPn/2 if K≥3/n, and if the normal connection of M is semi-flat.

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Li, Nan. "Vanishing Theorems for p-Harmonic Forms on Submanifolds in Spheres." Journal of Contemporary Educational Research 8, no. 4 (2024): 48–53. http://dx.doi.org/10.26689/jcer.v8i4.6671.

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In this paper, we present some vanishing theorems for p-harmonic forms on -super stable complete submanifold M immersed in sphere Sn + m. When 2 ≤ 1 ≤ n-2, M has a flat normal bundle. Assuming that M is a minimal submanifold and δ > 1(n–1)p2 / 4n[p–1+(p–1)2kp], we prove a vanishing theorem for p-harmonic ℓ-forms.

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Mahi, F., and M. Belkhelfa. "Legendrian normally flat submanifols of $\mathcal{S}$-space forms." Carpathian Mathematical Publications 12, no. 1 (2020): 69–78. http://dx.doi.org/10.15330/cmp.12.1.69-78.

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In the present study, we consider a Legendrian normally flat submanifold $M$ of $(2n+s)$-dimensional $\mathcal{S}$-space form $\widetilde{M}^{2n+s}(c)$ of constant $\varphi$-sectional curvature $c$. We have shown that if $M$ is pseudo-parallel then $M$ is semi-parallel or totally geodesic. We also prove that if $M$ is Ricci generalized pseudo-parallel, then either it is minimal or $L=\frac{1}{n-1}$, when $c\neq -3s$.

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Loo, Tee-How. "Sasakian anti-holomorphic submanifolds of a Kaehler manifold." Tamkang Journal of Mathematics 31, no. 4 (2000): 297–304. http://dx.doi.org/10.5556/j.tkjm.31.2000.387.

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Let $ M$ be a connected Sasakian anti-holomorphic submanifold of a Kaehler mnifold with flat norml connection and with dim $ D\ge 4$, where $ D$ is the holomorphic distribution on $ M$. We show that $ M$ is locally Riemannian product $ M' \times M''$ where $ M'$ is homothetic to a Sasakian manifold and $ M''$ is a locally Euclidean space.

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15

Ok Bayrakdar, Z., and T. Bayrakdar. "Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations." Advances in Mathematical Physics 2018 (2018): 1–8. http://dx.doi.org/10.1155/2018/7590847.

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We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers’ equations describe surfaces attached to an ODE of the form dx/dt=u(t,x) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the in

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16

CANARUTTO, DANIEL. "QUANTUM BUNDLES AND QUANTUM INTERACTIONS." International Journal of Geometric Methods in Modern Physics 02, no. 05 (2005): 895–917. http://dx.doi.org/10.1142/s0219887805000855.

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A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles ("quantum bundles") over the space–time manifold yield a quantum "formalism" along any 1-dimensional timelike submanifold (a "detector"); in the flat, inertial case this reproduces the basic results of the usual quantum field theory, while in general it could be seen as a local, "linearized" description of the actual physics.

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Smoczyk, Knut, Mao-Pei Tsui, and Mu-Tao Wang. "Generalized Lagrangian mean curvature flows: The cotangent bundle case." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 750 (2019): 97–121. http://dx.doi.org/10.1515/crelle-2016-0047.

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Abstract In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian su

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18

ISSE, TOSHIKI. "UNIFICATION OF GRAVITY, GAUGE AND HIGGS FIELDS BY CONFINED QUANTUM FIELDS II-EFFECTIVE THEORY." Modern Physics Letters A 08, no. 10 (1993): 947–60. http://dx.doi.org/10.1142/s0217732393000970.

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Dynamics of quantized free fields (of spin 0 and 1/2) contained in a subspace V* of an (N+4)-dimensional flat space V is studied. The space V* is considered as a neighborhood of a four-dimensional submanifold M arbitrarily embedded into V. We show that Einstein SO (N)-Yang-Mills Higgs theory is induced as a low energy effective theory of the system. Gravity, SO (N) gauge fields and Higgs fields are obtained from embedding functions of M.

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19

Avila, Artur, Alex Eskin, and Martin Möller. "Symplectic and isometric SL(2,#x211D;)-invariant subbundles of the Hodge bundle." Journal für die reine und angewandte Mathematik (Crelles Journal) 2017, no. 732 (2017): 1–20. http://dx.doi.org/10.1515/crelle-2014-0142.

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Abstract Suppose N is an affine {\mathrm{SL}(2,{\mathbb{R}})} -invariant submanifold of the moduli space of pairs (M,\omega) , where M is a curve, and ω is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal {\mathrm{SL}(2,{\mathbb{R}})} -invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N. As a corollary, it follows that the Hodge bundle of N is semisimple.

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20

KOIKE, TAKAYUKI. "HIGHER CODIMENSIONAL UEDA THEORY FOR A COMPACT SUBMANIFOLD WITH UNITARY FLAT NORMAL BUNDLE." Nagoya Mathematical Journal 238 (June 13, 2018): 104–36. http://dx.doi.org/10.1017/nmj.2018.22.

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Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher codimensional generalization of Ueda’s result, we give a sufficient condition for the existence of a nonsingular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semipositive curvature on a nef line bundle.

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Doubrov, Boris, Evgeny V. Ferapontov, Boris Kruglikov, and Vladimir S. Novikov. "Integrable Systems in Four Dimensions Associated with Six-Folds in Gr(4, 6)." International Mathematics Research Notices 2019, no. 21 (2018): 6585–613. http://dx.doi.org/10.1093/imrn/rnx308.

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Abstract Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V. A submanifold X ⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V whose Gaussian image is contained in X. We investigate a special case of this construction where X is a six-fold in Gr(4, 6). The corresponding system Σ(X) reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems Σ(X). Thes

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22

Wiliardy, Abednego, and Bobby Eka Gunara. "Higher dimensional maximally symmetric stationary manifold with pure gauge condition and codimension one flat submanifold." Journal of Physics: Conference Series 739 (August 2016): 012005. http://dx.doi.org/10.1088/1742-6596/739/1/012005.

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23

Canevari, Giacomo, and Giandomenico Orlandi. "Topological Singular Set of Vector-Valued Maps, II: $$\varGamma $$-convergence for Ginzburg–Landau type functionals." Archive for Rational Mechanics and Analysis 241, no. 2 (2021): 1065–135. http://dx.doi.org/10.1007/s00205-021-01671-2.

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AbstractWe prove a $$\varGamma $$ Γ -convergence result for a class of Ginzburg–Landau type functionals with $${\mathscr {N}}$$ N -well potentials, where $${\mathscr {N}}$$ N is a closed and $$(k-2)$$ ( k - 2 ) -connected submanifold of $${\mathbb {R}}^m$$ R m , in arbitrary dimension. This class includes, for instance, the Landau-de Gennes free energy for nematic liquid crystals. The energy density of minimisers, subject to Dirichlet boundary conditions, converges to a generalised surface (more precisely, a flat chain with coefficients in $$\pi _{k-1}({\mathscr {N}})$$ π k - 1 ( N ) ) which s

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ISSE, TOSHIKI. "UNIFICATION OF GRAVITY, GAUGE AND HIGGS FIELDS BY CONFINED QUANTUM FIELDS – MATHEMATICAL FORMULATION." Modern Physics Letters A 08, no. 09 (1993): 837–49. http://dx.doi.org/10.1142/s0217732393000878.

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Dynamics of quantized free fields (of spin 0 and 1/2) contained in a subspace V* of an (N+4)-dimensional flat space V is studied. The space V* is considered as a neighborhood of a four-dimensional submanifold M arbitrarily embedded into V. We study the system as a simple model of unified theory of gravity (g), SO (N) gauge fields (A) and Higgs fields (ɸ). In this paper classical treatment of the system is given. We show that, especially when the fields have spin 1/2, the system is described by an infinite number of fields in interacting with g, A and ɸ. The fields g, A and ɸ are induced themse

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Alho, Artur, та Claes Uggla. "Quintessential α-attractor inflation: a dynamical systems analysis". Journal of Cosmology and Astroparticle Physics 2023, № 11 (2023): 083. http://dx.doi.org/10.1088/1475-7516/2023/11/083.

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Abstract The equations for quintessential α-attractor inflation with a single scalar field, radiation and matter in a spatially flat FLRW spacetime are recast into a regular dynamical system on a compact state space. This enables a complete description of the solution space of these models. The inflationary attractor solution is shown to correspond to the unstable center manifold of a de Sitter fixed point, and we describe connections between slow-roll and dynamical systems approximations for this solution, including Padé approximants. We also introduce a new method for systematically obtainin

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Shugailo, O. O. "Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping." Matematychni Studii 60, no. 1 (2023): 99–112. http://dx.doi.org/10.30970/ms.60.1.99-112.

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In the paper we study equiaffine immersions $f\colon (M^n,\nabla) \rightarrow {\mathbb{R}}^{n+2}$ with flat connection $\nabla$ and one-dimensional Weingarten mapping. For such immersions there are two types of the transversal distribution equiaffine frame.We give a parametrization of a submanifold with the given properties for both types of equiaffine frame. The main result of the paper is contained in Theorems 1, 2 and Corollary 1: Let $f\colon ({M}^n,\nabla)\rightarrow({\mathbb{R}}^{n+2},D)$ be an affine immersion with pointwise codimension 2, equiaffine structure, flat connection $\nabla$,

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ANIELLO, P., J. CLEMENTE-GALLARDO, G. MARMO, and G. F. VOLKERT. "CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT I: PURE STATES." International Journal of Geometric Methods in Modern Physics 07, no. 03 (2010): 485–503. http://dx.doi.org/10.1142/s0219887810004300.

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The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Rieman

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Ferapontov, E. V. "Dirac reduction of the hamiltonian operator? IJd/dx to a submanifold of euclidean space with flat normal connection." Functional Analysis and Its Applications 26, no. 4 (1992): 298–300. http://dx.doi.org/10.1007/bf01075056.

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Wu, B. Y. "Some results on Finsler submanifolds." International Journal of Mathematics 27, no. 03 (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

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Morvan, Jean-Marie, and Georges Zafindratafa. "Conformally flat submanifolds." Annales de la faculté des sciences de Toulouse Mathématiques 8, no. 3 (1987): 331–48. http://dx.doi.org/10.5802/afst.643.

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Özgür, Cihan. "Submanifolds satisfying some curvature conditions imposed on the Weyl tensor." Bulletin of the Australian Mathematical Society 67, no. 1 (2003): 95–101. http://dx.doi.org/10.1017/s0004972700033554.

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In this paper we define Weyl semiparallel ( · h = 0) and Weyl 2-semiparallel ( · ∇̅h = 0) submanifolds. We consider n-dimensional normally flat submanifolds satisfying these curvature conditions in (n + d)-dimensional Euclidean space n+d. We also consider normally flat submanifolds in (n + d)-dimensional Euclidean space n+d satisfying the condition  · h = LQ (g, h) and  · ∇̅h = LQ(g, ∇̅h).

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Zhang, Jun. "Divergence Function, Duality, and Convex Analysis." Neural Computation 16, no. 1 (2004): 159–95. http://dx.doi.org/10.1162/08997660460734047.

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From a smooth, strictly convex function Φ: Rn → R, a parametric family of divergence function DΦ(α) may be introduced: [Formula: see text] for x, y, ε int dom(Φ) and for α ε R, with DΦ(±1 defined through taking the limit of α. Each member is shown to induce an α-independent Riemannian metric, as well as a pair of dual α-connections, which are generally nonflat, except for α = ±1. In the latter case, D(±1)Φ reduces to the (nonparametric) Bregman divergence, which is representable using and its convex conjugate Φ * and becomes the canonical divergence for dually flat spaces (Amari, 1982, 1985; A

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Dajczer, Marcos, and Luis A. Florit. "On conformally flat submanifolds." Communications in Analysis and Geometry 4, no. 2 (1996): 261–84. http://dx.doi.org/10.4310/cag.1996.v4.n2.a3.

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Dajczer, M., C. R. Onti, and Th Vlachos. "Conformally flat submanifolds with flat normal bundle." manuscripta mathematica 163, no. 3-4 (2019): 407–26. http://dx.doi.org/10.1007/s00229-019-01158-1.

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Öztürk, Sermin, та Hakan Öztürk. "Certain Class of Almost α -Cosymplectic Manifolds". Journal of Mathematics 2021 (22 грудня 2021): 1–9. http://dx.doi.org/10.1155/2021/9277175.

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This paper concerned with almost α -cosymplectic manifolds satisfying conformally flat condition. Firstly, we investigate Kaehler integral submanifolds of almost α -cosymplectic manifolds. Next, we study conformally flat almost α -cosymplectic manifolds of dim ≥ 5 whose integral submanifolds are Kaehler. Finally, an illustrative example is constructed to verify our result.

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Yadav, Sunıl, Abdul Haseeb, and Ahmet Yıldız. "Conformal $\eta$-Ricci-Yamabe solitons on submanifolds of an $(LCS)_n$-manifold admitting a quarter-symmetric metric connection." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 73, no. 3 (2024): 611–29. http://dx.doi.org/10.31801/cfsuasmas.1382928.

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This paper presents some results for conformal $\eta$-Ricci-Yamabe solitons (CERYS) on invariant and anti-invariant submanifolds of a $(LCS)_n$-manifold admitting a quarter-symmetric metric connection (QSMC). In addition, we developed the characterization of CERYS on $M$-projectively flat, $Q$-flat, and concircularly flat anti-invariant submanifolds of a $(LCS)_n$-manifold with respect to the aforementioned connection. Finally, we construct an example that appoints some of our inference.

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Petrovic-Torgasev, Miroslava, та Anica Pantic. "Some properties of δ(2,2) Chen ideal submanifolds". Filomat 31, № 7 (2017): 2163–66. http://dx.doi.org/10.2298/fil1707163p.

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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 (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 t

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Dajczer, Marcos, and Ruy Tojeiro. "Submanifolds with Nonparallel First Normal Bundle." Canadian Mathematical Bulletin 37, no. 3 (1994): 330–37. http://dx.doi.org/10.4153/cmb-1994-049-3.

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AbstractWe provide a complete local geometric description of submanifolds of spaces with constant sectional curvature where the first normal spaces, that is, the subspaces spanned by the second fundamental form, form a vector subbundle of the normal bundle of low rank which is nonparallel in the normal connection. We also characterize flat submanifolds with flat normal bundle in Euclidean space satisfying the helix property.

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Matzeu, Paola. "Submanifolds of Weyl Flat Manifolds." Monatshefte f?r Mathematik 136, no. 4 (2002): 297–311. http://dx.doi.org/10.1007/s00605-002-0477-0.

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Verstraelen, Leopold, and Georges Zafindratafa. "On normally flat Einstein submanifolds." International Journal of Mathematics and Mathematical Sciences 20, no. 3 (1997): 497–501. http://dx.doi.org/10.1155/s0161171297000677.

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42

Terng, Chuu-Lian. "Submanifolds with flat normal bundle." Mathematische Annalen 277, no. 1 (1987): 95–111. http://dx.doi.org/10.1007/bf01457280.

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43

Fu, Hai-Ping. "Minimal submanifolds with flat normal bundle." Kodai Mathematical Journal 33, no. 2 (2010): 211–16. http://dx.doi.org/10.2996/kmj/1278076337.

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Haller, Stefan, and Cornelia Vizman. "Nonlinear flag manifolds as coadjoint orbits." Annals of Global Analysis and Geometry 58, no. 4 (2020): 385–413. http://dx.doi.org/10.1007/s10455-020-09725-6.

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Abstract A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Fréchet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application, we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags.

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DAJCZER, MARCOS, and RUY TOJEIRO. "AN EXTENSION OF THE CLASSICAL RIBAUCOUR TRANSFORMATION." Proceedings of the London Mathematical Society 85, no. 1 (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

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Matsuo, Koji. "Pseudo-Bochner-flat locally conformal Kähler submanifolds." Colloquium Mathematicum 108, no. 2 (2007): 305–15. http://dx.doi.org/10.4064/cm108-2-12.

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Dajczer, Marcos, and Ruy Tojeiro. "The Ribaucour transformation for flat Lagrangian submanifolds." Journal of Geometric Analysis 10, no. 2 (2000): 269–80. http://dx.doi.org/10.1007/bf02921825.

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Zuo, Dafeng, Qing Chen, and Yi Cheng. "Flat time-like submanifolds inS2n 12q(1)." Journal of Physics A: Mathematical and General 35, no. 47 (2002): 10197–204. http://dx.doi.org/10.1088/0305-4470/35/47/316.

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Romero-Fuster, M. C., and E. Sanabria-Codesal. "On the flat ridges of submanifolds of codimension 2 in Rn." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 4 (2002): 975–84. http://dx.doi.org/10.1017/s0308210500001967.

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We introduce the concept of flat ridges for submanifolds of codimension 2 from the viewpoint of their contact with hyperplanes. We characterize them geometrically, studying some of their properties. In particular, we see that the highest-order flat ridges coincide with the flattenings of the asymptotic lines and from this we obtain some lower bounds for their numbers under appropriate conditions.

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Mirzoyan, V. A., and G. S. Machkalyan. "Normally flat Ric-semisymmetric submanifolds in Euclidean spaces." Russian Mathematics 56, no. 9 (2012): 14–24. http://dx.doi.org/10.3103/s1066369x12090034.

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