A. A. Borisenko, V. O. Gorkavyy
Three-dimensional pseudo-spherical submanifolds in $\mathbb R^5$, whose Bianchi transformations are degenerate of rank 2, are studied. A complete description of such submanifolds is obtained in the case where the Bianchi transformations are holonomically degenerate.
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An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space $P^N (\mathbf{C})$. By means of the focal images, three basic types of submanifolds are discovered: cones, tangentially degenerate hypersurfaces, and torsal submanifolds. Moreover, for tangentially degenerate submanifolds, a structural theorem is proven. By this theorem, tangentially degenerate submanifolds that do not…
We obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds of constant sectional curvature of space forms, which we call the $L$-transformation. It allows to construct a family of such submanifolds starting with a given one and a vector-valued solution of a system of linear partial differential equations. We prove a decomposition theorem for the $L$-transformation, which is a far-reaching generalization of the classical permutability formula for the Ribaucour transformation of surfaces of constant curvature in Euclidean three space. As a consequenc…
We investigate half-lightlike submanifolds with planar normal sections of four dimensional pseudo Euclidean space. We obtain necessary and sufficient conditions for a half-lightlike submanifold of $R_{2}^{4}$ such that it has degenerate or non-degenerate planar normal sections.
We discuss results for the Ribaucour transformation of curves or of higher dimensional smooth and discrete submanifolds. In particular, a result for the reduction of the ambient dimension of a submanifold is proved and the notion of Ribaucour coordinates is derived using a Bianchi permutability result. Further, we discuss smoothing of semi-discrete curvature line nets and an interpolation by Ribaucour transformations.
We prove that every non-degenerate minimal submanifold of codimension two can be obtained as the energy concentration set of a family of critical maps for the (rescaled) Ginzburg-Landau functional. The proof is purely variational, and follows the strategy laid out by Jerrard and Sternberg, extending a recent result for geodesics by Colinet-Jerrard-Sternberg. The same proof applies also to the $U(1)$-Yang-Mills-Higgs and to the Allen-Cahn-Hilliard energies.
In this paper we show that two Kähler manifolds which do not share a Kähler submanifold, do not share either a Levi degenerate CR-submanifold with constant dimension Levi kernel. In particular, they do not share a CR-product. Further, we obtain that a Levi degenerate CR-submanifold of $\mathbb C^n$ cannot be isometrically immersed into a flag manifold.
$P$-categories and functions with degenerate singular submanifolds
研究流形上临界点集为同胚于同一流形P的子流形的不交并的光滑函数，给出存在具有最少此类子流形的函数的充分条件。
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds X^3 of rank 2. The authors study focal properties of these submanifolds and prove that they represent examples of different types of tangentially degenerate submanifolds. Namely, the submanifold X^3, corresponding in RP^5 to a smooth line γof the projective plane C, does not have…
Degenerate submanifolds of pseudo-Riemannian manifolds are quite difficult to study because there is no prefered connection when the submanifold is not totally geodesic. For the particular case of degenerate totally umbilical hypersurfaces, we show that there are "Weyl" connections adapted to the induced structure on the hypersurface. We begin the study of these with their holonomy.
The authors establish a relation of the theory of varieties with degenerate Gauss maps in projective spaces with the theory of congruences and pseudocongruences of subspaces and show how these two theories can be applied to the construction of induced connections on submanifolds of projective spaces and other spaces endowed with a projective structure.
Conformal-projective transformations on statistical and semi-Weyl manifolds with torsion
该研究探讨带挠率的统计流形与半Weyl流形上的共形射影变换，证明相关结构的不变性、子流形诱导结构的等价性、脐点的保性，分析类光与非退化超曲面的异同，并给出半Weyl流形的仿射分布实现方式。
Given a domain $Ω$ of $\mathbb{R}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\pa Ω$ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial Ω$ perpendicularly.
Normal Holonomy of Complex Hyperbolic Submanifolds
该研究证明复双曲空间$\mathbb{C}H^{n}$中相对零化指数为零的凯勒子流形的限制法曲率群始终可迁，通过提升至伪黎曼空间$\mathbb{C}^{n,1}$及新工具解决退化问题，对比$\mathbb{C}P^{n}$的Berger型结果并推进不定符号空间子流形几何研究。
Morse-Bott Volume Forms
本文研究莫尔斯-博特体积形式的微分同胚等价性，证明相对上同调类一致时两形式微分同胚，余维数≥2的零子流形情形下等价于总体积相等，并适配莫泽技巧到该场景。
We show that the regularized Riesz $\a$-energy for closed submanifolds $M$ in $\RR^n$ blows up as $M$ degenerates to have double points if $\a\le-2\dim M$. This gives theoretical foundation of numerical experiments to evolve surfaces to decrease the energy which have been carried out since 90's.
Finding examples of tangentially degenerate submanifolds (submanifolds with degenerate Gauss mappings) in an Euclidean space $R^4$ that are noncylindrical and without singularities is an important problem of differential geometry. The first example of such a hypersurface was constructed by Sacksteder in 1960. In 1995 Wu published an example of a noncylindrical tangentially degenerate algebraic hypersurface in $R^4$ whose Gauss mapping is of rank 2 and which is also without singularities. This example was constructed (but not published) by Bourgain. In this paper, the authors analyze Bourgain's…
Isotropic submanifolds of $T\mathbb{S}^n$ and their focal sets
研究$T\mathbb{S}^n$上各向同性子流形及其焦点集，揭示其与超曲面曲率的关系。
Causal Invariance of the Contact Number of Pseudo-Riemannian Submanifolds
该研究证明伪黎曼子流形的普通、类空、类时接触数等价且重合，构造了接触数均为6的洛伦兹曲面，将无穷接触极限纳入半黎曼理论。
The space of all non degenerate bilinear structures on a manifold $M$ carries a one parameter family of pseudo Riemannian metrics. We determine the geodesic equation, covariant derivative, curvature, and we solve the geodesic equation explicitly. Each space of pseudo Riemannian metrics with fixed signature is a geodesically closed submanifold. The space of non degenerate 2-forms is also a geodesically closed submanifold. Then we show that, if we fix a distribution on $M$, the space of all Riemannia metrics splits as the product of three spaces which are everywhere mutually orthogonal, for the …
Let $(\MM ,{\tilde g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation $$ ε^2Δ_{ {\tilde g}} {u}\,+\, (1 - {u}^2)u \,=\,0\quad \mbox{in } \MM, $$ where $ε$ is a small parameter. Let $\KK\subset \MM$ be an $(N-1)$-dimensional smooth minimal submanifold that separates $\MM$ into two disjoint components. Assume that $\KK$ is non-degenerate in the sense that it does not support non-trivial Jacobi fields, and that $|A_{\KK}|^2+\mbox{Ric}_{\tilde g}(ν_{\KK}, ν_{\KK})$ is positive along $\KK$. Then for each integer $m\geq 2$, we esta…
Maps from Riemannian manifolds into non-degenerate Euclidean cones
本文研究连通非紧黎曼流形到非退化欧氏锥的光滑映射，基于能量、张力与度量参数给出锥宽下界，并推广了调和映射、极小浸入与 Kähler 子流形的经典结论及等距浸入的排除性结果。
We associate a natural $λ$-family ($λ\in \R \setminus \{0\} $) of flat Lagrangian immersions in $\C^n$ with non-degenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same Lax pair as the first order integrable system associated to the symmetric space $\frac{\U(n) \ltimes \C^n}{\OO(n) \ltimes \R^n}$. An interesting observation is that the family degenerates to an Egoroff net on $\R^n$ when $λ\to 0$. We construct an action of a rational loop group on such immersions by identifying its generators and computing their dressing actions. The ac…
The half-space model of pseudo-hyperbolic space
本文为任意满足 p≥1 的伪双曲空间 H^{p,q} 构造了半空间模型，并刻画了其中的测地线、全测地子流形、极限球面、等距群及无穷远边界。
Let $Gr(d,n)$ be the Grassmannian of $d$-dimensional linear subspaces of an $n$-dimensional vector space $V$. A submanifold $X\subset Gr(d, n)$ 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 sixfold 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 …
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