we study the notion of harmonicity in the sense of symplectic geometry, and investigate the geometric properties of harmonic thom forms and distributional thom currents, dual to different types of submanifolds. we show that the harmonic thom form associated to a symplectic submanifold is nowhere vanishing. we also construct symplectic smoothing operators which preserve the harmonicity of distri...

Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G2-manif...

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.

We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study index for a vector field with non-isolated singularities on a submanifold. As an application, our studies give conceptual proofs of a classical result of Chern. The objective of this paper is to define, study and use the secondary Chern-Euler class for a ...

of the Dissertation Gluing Techniques in Calibrated Geometry by Yongsheng Zhang Doctor of Philosophy in Mathematics Stony Brook University 2013 This thesis is concerned with the question: given a submanifold (perhaps with singularities), when is it possible to change the metric in some specific way so that the submanifold becomes homologically mass-minimizing? We studied this question for “hori...

Let ƒ: N —• M be a C°° map of oriented compact manifolds, and let L be an oriented closed submanifold of codimension q > 1 in M. If w is a closed form Poincaré dual to L, we show that f~L, with multiplicities counted, is Poincaré dual to ƒ *w in N and is even meaningful on a "secondary" level. This leads to generalized versions of the Hopf invariant, the Hopf index theorem and the Bezout theore...

Let M be an n-dimensional submanifold in the simply connected space form F(c) with c + H > 0, where H is the mean curvature of M . We verify that if M(n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies RicM ≥ (n−2)(c+H2), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n+1)-sphere with n = even, or CP ( 4 3 (c+H )) ...

We show that any closed oriented immersed isotropic minimal surface Σ with genus gΣ in S ⊂ C is (1) Legendrian (and totally geodesic) if gΣ = 0; (2) either Legendrian or with exactly 2gΣ − 2 Legendrian points if gΣ ≥ 1. In general, any compact oriented immersed isotropic minimal submanifold L ⊂ S ⊂ C must be Legendrian if its first Betti number is zero. Corresponding results for nonorientable l...

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 differential invariants of an associated manifold under Lie group action, and therefore signature submanifold is a key for solving equivalence problems.

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an oriented connected closed π1-injectively immersed quasiFuchsian subsurface. Second, every rationally null-homologous, π1-injectively immersed oriented closed 1-sub...