New construction of special Lagrangian submanifolds in <Emphasis Type=Italic>T</Emphasis>*R<Superscript><Emphasis Type=Italic>n</Emphasis></Superscript> equipped with the standard metric

A class of twisted special Lagrangian submanifolds in T＊R^n and a kind of austere submanifold from conormal bundle of minimal surface of R^3 are constructed.     

On almost contact metric hypersurfaces with type number 1 in 6-dimensional Kähler submanifolds of the Cayley algebra

We prove that the condition for the type number of an almost contact metric surface in a six-dimensional Kähler submanifold of the Cayley algebra to be 0 or 1 is not only necessary but also sufficient condition for the almost contact metric structure to be cosymplectic.     

The Dichotomy Between Traces on d-sets F in R^n and the Density of D（R^n／Г） in Function Spaces

A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.     

On the Geometrical Meaning of the Dynamic Equations in Analytical Mechanics

In this paper,the modern geometrical structure of analytical mechanics,the exteriordifferential forms and the geometrical meaning of dynamic equations are briefly discussed.     

Some fixed point results for relation theoretic weak $$varphi $$ φ -contractions in cone metric spaces equipped with a binary relation and application to the system of Volterra type equations

Positivity - In this paper, some fixed point results for relation theoretic weak $$varphi $$ -contractions on cone metric spaces which is defined over a Banach algebra and equipped with a binary...     

Precise asymptotics in Chung’s law of the iterated logarithm

Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.     

Deformation of special Lagrangian suborbifolds

The purpose of this paper is to generalize the deformation theory of special Lagrangian submanifolds developed by Mclean and Hitchin to special Lagrangian suborbifolds.     

Affine geometry of special real submanifolds of Cn

In this paper we propose an affine analogue and generalization of the geometry of special Lagrangian submanifolds of Cⁿ.      

Evolution Equations for Special Lagrangian 3-Folds in C3

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C ^m . The previous paper (Math. Ann. 320 (2001), 757–797), defined the idea of evolution data, which includes an (m – 1)-submanifold P in R ⁿ , and constructed a family of special Lagrangian m-folds N in C ^m , which are swept out by the image of P under a 1-parameter family of affine maps _t : R ⁿ C ^m , satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C³. We find a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C³.Our main results are a number of new families of special Lagrangian 3-foldsin C³, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R³ or ¹× R². Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind.We hope these 3-folds will be helpful in understanding singularities ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.     

An optimal rigidity theorem for complete submanifolds in a sphere

It is proved that if M^n is an n-dimensional complete submanifold with parallel mean curvature vector and flat normal bundle in S^n＋p（1）, and if supM S 〈 α（n, H）, where α（n,H）=n＋n^3/2（n-1）H^2-n（n-2）/n（n-1）√n^2H^4＋4（n-1）H^2,then M^n must be the totally urnbilical sphere S^n（1/√1＋H^2）.An example to show that the pinching constant α（n, H） appears optimal is given.     

Optimal general inequalities for Lagrangian submanifolds in complex space forms

Lagrangian submanifolds appear naturally in the context of classical mechanics. Moreover, they play some important roles in supersymmetric field theories as well as in string theory. In this paper we establish general inequalities for Lagrangian submanifolds in complex space forms. We also provide examples showing that these inequalities are the best possible. Moreover, we provide simple non-minimal examples which satisfy the equality case of the improved inequalities.     

Special Lagrangian Submanifolds with Isolated Conical Singularities. I. Regularity

This is the first in a series of five papers studying special Lagrangian submanifolds(SLV m-folds) X in almost Calabi–Yau m-folds M with singularitiesx ₁, ..., x _n locally modelled on special Lagrangiancones C ₁, ..., C _n in ^m with isolated singularities at 0. Readers are advised to begin with Paper V.This paper lays the foundations for the series, giving definitions and provingauxiliary results in symplectic geometry and asymptotic analysis that will be needed later.It also proves results on the regularity of X near its singular points.We show that X converges to the cone C _i near x _i with all its derivatives,at rates determined by the eigenvalues of the Laplacian on C _i .We show that if X is a special Lagrangian integral current with a tangent cone C at x satisfying some conditions, then X has an isolated conical singularity at x in our sense. We also prove analogues of many of our results for Asymptotically Conical SL m-folds in ^m .     

Universal Maslov class of a Bohr-Sommerfeld Lagrangian embedding into a pseudo-Einstein manifold

We show that in the case of a Bohr-Sommerfeld Lagrangian embedding into a pseudo-Einstein symplectic manifold, a certain universal 1-cohomology class, analogous to the Maslov class, can be defined. In contrast to the Maslov index, the presented class is directly related to the minimality problem for Lagrangian submanifolds if the ambient pseudo-Einstein manifold admits a Kähler-Einstein metric. We interpret the presented class geometrically as a certain obstruction to the continuation of one-dimensional supercycles from the Lagrangian submanifold to the ambient symplectic manifold.     

Theory of submanifolds,associativity equations in 2D topological quantum field theories,and Frobenius manifolds

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of flat torsionless potential submanifolds. We show that all flat torsionless potential submanifolds in pseudo-Euclidean spaces bear natural structures of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (in fact, the structural constants are given by the set of the Weingarten operators of the submanifolds). We prove that each N-dimensional Frobenius manifold can be locally represented as a flat torsionless potential submanifold in a 2N-dimensional pseudo-Euclidean space. By our construction, this submanifold is uniquely determined up to motions. Moreover, we consider a nonlinear system that is a natural generalization of the associativity equations, namely, the system describing all flat torsionless submanifolds in pseudo-Euclidean spaces, and prove that this system is integrable by the inverse scattering method. To the memory of my wonderful mother Maya Nikolayevna Mokhova (4 May 1926–12 September 2006) Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 368–376, August, 2007.