We discuss the behaviour of vertices and inflexions of one-parameter families of plane curves which include a singular member. These arise a sectons of smooth surfaces by families of planes parallel to the tangent plane at a given point. We cover all the generic cases, namely elliptic, umbilic, hyperbolic, parabolic and cusp of Gauss points. This work is preliminary to an investigation of symme...

We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These results extend to higher dimensions earlier three dimensional work of J. C. C. N...

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

In this short note, we discuss certain examples of Legendrian submanifolds, whose linearized contact (co)homology groups over integers have non-vanishing algebraic torsion. More precisely, for a given arbitrary finitely generated abelian group [Formula: see text] and positive integer text], construct submanifolds the standard vector space text]th computed with respect to augmentation is isomorp...

For a pair of smooth transversally intersecting submanifolds in some enveloping manifold, we study the algebra generated by pseudodifferential operators and (co)boundary corresponding to submanifolds. We establish that such an has 18 types generating elements. from this algebra, define concept symbol obtain composition formula.

this article concerned on the study of signature submanifolds for curves under lie group actions se(2), sa(2) and for surfaces under se(3). signature submanifold is a regular submanifold which its coordinate components are dierential invariants of an associated manifold under lie group action, and therefore signature submanifold is a key for solving equivalence problems.

By a theorem of Mclean, the deformation space of an associative submanifold Y of an integrable G2-manifold (M, ϕ) can be identified with the kernel of a Dirac operator D / : Ω 0 (ν) → Ω 0 (ν) on the normal bundle ν of Y. Here, we generalize this to the non-integrable case, and also show that the deformation space becomes smooth after perturbing it by natural parameters, which corresponds to mov...

Associated to oriented surfaces immersed in R 3 , here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature K, given by the product of the principal curvatures k1, k2 of the immersion, is positive. The leaves of the foliations are the lines of M-mean curvature, along which the normal curvature of the immersion is given by a...

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

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M . Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two application...