Alexander Yampolsky
We give a full description of totally geodesic submanifolds in the tangent bundle of a Riemannian 2-manifold of constant curvature and present a new class of a cylinder-type totally geodesic submanifolds in the general case.
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We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of $(κ,μ)$-spaces with Boeckx invariant $I \leq -1$. In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct.
In Kaehler manifolds are investigated conformally flat totally real submanifolds, which are semiparallel or have semiparallel mean curvature vector.
The present paper deals with the study of totally real submanifolds and $\textit{C}$-totally real submanifolds of $(LCS)_n$-manifolds with respect to Levi-Civita connection as well as quarter symmetric metric connection. It is proved that scalar curvature of $\textit{C}$-totally real submanifolds of $(LCS)_n$-manifold with respect to both the said connections are same.
The purpose of this paper is to classify totally umbilical slant submanifolds of a Kenmotsu manifold. We prove that a totally umbilical slant submanifold $M$ of a Kenmotsu manifold $\bar M$ is either invariant or anti-invariant or $dim M=1$ or the mean curvature vector $H$ of $M$ lies in the invariant normal subbundle. Moreover, we find with an example that every totally umbilical proper slant submanifold is totally geodesic.
The present paper is a continuation of our previous work, where a class of half lightlike submanifolds of almost contact B-metric manifolds was introduced. We study curvature properties of totally and screen totally umbilical such submanifolds as well as of the corresponding semi-Riemannian submanifolds with respect to the associated B-metric.
Two geometric inequalities are established for Einstein totally real submanifolds in a complex space form. As immediate applications of these inequalities, some non-existence results are obtained.
Hopf fibrations and totally geodesic submanifolds
本文分类 Hopf-Berger 球面中的全测地子流形，并揭示仅在正曲率情形出现的实射影空间、等距非全等及非外蕴齐性等特殊例子。
In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces. In this way a classification of the totally geodesic submanifolds in the complex quadric $Q^m := \SO(m+2)/(\SO(2) \times \SO(m))$ is obtained. It turns out that the earlier classification of totally geodesic submanifolds of $Q^m$ by Chen and Nagano is incom…
Totally geodesic submanifolds in products of rank one symmetric spaces
本文分类了秩一对称空间任意乘积中的全测地子流形，并在 Hermitian 对称空间中给出了无穷多个具有非平凡常 Kähler 角的不可约全测地子流形实例。
We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We prove a result analogous to the index conjecture: for every irreducible symmetric space of non-compact type, there exists a totally geodesic submanifold which is of minimal codimension and whose non-flat irreducible factors have Dynkin index equal to one.
A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which is characterized by the condition that the second fundamental form is proportional to the metric. It is also a generalization of the notion of a totally geodesic submanifold. In this paper, we classify congruent classes of full totally umbilical submanifolds in non-flat pseudo-Riemannian space forms, and consider their moduli spaces. As a consequence, we show that some moduli spaces of isometric immersions between space forms which one of the same constant curvature are non-Hausdorff.
In this paper, by studying the position of umbilical normal vectors in the normal bundle, we prove that pseudo-umbilical totally real submanifolds with flat normal connection in non-flat complex space forms must be minimal.
In the first part of this expository article, the most important constructions and classification results concerning totally geodesic submanifolds in Riemannian symmetric spaces are summarized. In the second part, I describe the results of my classification of the totally geodesic submanifolds in the Riemannian symmetric spaces of rank 2.
In this article, I classify the totally geodesic submanifolds in the complex 2-Grassmannians and in the quaternionic 2-Grassmannians. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank 2, published by Chen and Nagano (B.-Y. Chen, T. Nagano, "Totally geodesic submanifolds of symmetric spaces, II", Duke Math. J. 45 (1978), 405--425) is incomplete. For example, G_2(H^n) with n >= 7 contains totally geodesic submanifolds isometric to a HP^2, its metric scaled such that the minimal sectional curvatur…
In this paper I have studied about CR(Cauchy-Riemann)-submanifolds of Lorentzian Concircular Structure manifold ((LCS)n-manifold), Lorentzian Para-Sasakian(LP)-cosymplectic manifold, S-manifold and Generalized Kenmotsu (GKM) manifold. I have discussed some results regarding distribution, structure vector field, totally geodesic submanifold, leaf etc.. I have obtained results on totally umbilical contact CR-submanifold where the anti-invariant distribution has some properties. Next, I have studied some results about D-totally geodesic CR-submanifold (D is the distribution), a contact CR-submani…
We use the Cartan representations of $SO(3)$ and $SU(3)$, and an irreducible 14-dimensional representation of $Sp(3)$ to construct certain totally geodesic submanifolds in "skew" position in the complex quadrics, the complex 2-Grassmannians and the quaternionic 2-Grassmannians.
Totally geodesic submanifolds of the homogeneous nearly Kähler 6-manifolds and their G2-cones
本文分类齐性近凯勒6维流形及其G2锥的全测地子流形，开发相关新技术，还得到单连通齐性空间中带自交的全测地子流形的例子。
We introduce sequential warped product submanifolds of Kaehler manifolds, provide examples and establish Chen's inequality for such submanifolds. The equality case is also studied. Moreover, by inspiring Lawson and Simons's integral currrent's theorem on a submanifold, we find a similar pinching inequality for a sequential warped product submanifold and obtain geometric results when the equality case is satisfied.
In the paper we investigate submanifolds in a tangent bundle endowed with g-natural metric G, defined by a vector field on a base manifold. We give a sufficient condition for a vector field on M to defined totally geodesic submanifold in (TM,G). The parallel vector field is discussed in more detail.
The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a Kähler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the normal bundle and of constant length of the second fundamental form (or equivalently of constant scalar curvature) of a complex space form N is totally geodesic in N or of positive scalar curvature. Moreover, if the scalar curvature of M vanishes, then M is flat. Theorem 3. A comp…
We prove that a maximal totally complex submanifold $N^{2n}$ of the quaternionic projective space $\mathbb{H}\mathbb{P}^n$ ($n\geq 2$) is a parallel submanifold, provided one of the following conditions is satisfied: (1) $N$ is the orbit of a compact Lie group of isometries, (2) the restricted normal holonomy is a proper subgroup of ${\rm U}(n)$.
A convergence result for Mean Curvature Flow of totally real submanifolds
该研究在数学微分几何领域，针对精确量化意义下“几乎极小”的全实子流形，建立了其平均曲率流的收敛性结果，推广并有效化了H. Li的拉格朗日平均曲率流相关结论。
An example of non-compact totally complex submanifolds of compact quaternionic Kähler symmetric spaces
本文构造了紧四元数凯勒对称空间（四元数射影空间除外）的非紧极大维全复子流形示例，分析其结构与群作用性质，并证明了相关紧子流形的存在性限制。
In this paper, we obtain a basic Chen's inequality for a C-totally real submanifold in a generalized $(κ,μ)$-contact space forms involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvature on the right hand side. Inequalities between the squared mean curvature and Ricci curvature and between the squared mean curvature and $k$-Ricci curvature are also obtained.
We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequalities involving the Laplacian of the warping function and the squared mean curvature of $C$-totally real warped product submanifolds of $(κ,μ) $-space forms, Sasakian space forms and non-Sasakian $(κ,μ) $-manifolds. Then we obtain obstructions to the existence of minimal isometr…
For $n\geq 4$ we show that generic closed Riemannian $n$-manifolds have no nontrivial totally geodesic submanifolds, answering a question of Spivak. An immediate consequence is a severe restriction on the isometry group of a generic Riemannian metric. Both results are widely believed to be true, but we are not aware of any proofs in the literature.
We study the normal holonomy group, i.e. the holonomy group of the normal connection, of a CR-submanifold of a complex space form. We complete the local classification of normal holonomies for complex submanifolds. We show that the normal holonomy group of a coisotropic submanifold acts as the holonomy representation of a Riemannian symmetric space. In case of a totally real submanifold we give two results about reduction of codimension. We describe explicitly the action of the normal holonomy in the case in which the totally real submanifold is contained in a totally real totally geodesic sub…
We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic manifold is by itself harmonic. We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a particular radius; this completes the classification of Einstein hypersurf…
$H$-umbilicity was introduced as an analogue of total umbilicity for Lagrangian submanifolds since, in some relevant cases, totally umbilical Lagrangian submanifolds are automatically totally geodesic. In this paper, we show that in the homogeneous nearly Kähler $\mathbb{S}^3\times\mathbb{S}^3$, also $H$-umbilical Lagrangian submanifolds are automatically totally geodesic.
Partially Totally Real Submanifolds of Sasakian Manifolds
本文针对切于Reeb向量场的Sasakian流形子流形，发展了部分全实（PTR）子流形的对应框架，证明了几类特殊子流形属于该框架，研究了相关几何性质并给出了模型说明。
Rigidity and vanishing results on totally real submanifolds under $L^p$-integrable conditions
本文修正并推广了Cuong等人在复空间形式中全实极小子流形上的刚性与消失性结果，主要贡献是将$L^p$可积条件中$p$的取值范围进行了扩展。
Totally geodesic submanifolds and polar actions on Stiefel manifolds
本文分类了正交二标架实Stiefel流形的全测地子流形及其上的极作用，证明极作用轨道为Grassmannian上极作用的提升，并完成余齐一情形下实、复、四元数Stiefel流形的分类。
We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].
Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equality holds, that is, rk(M) = i(M). In this context we also obtain an explicit classification of all non-semisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric …
It is well-known that if $ξ$ is a smooth vector field on a given Riemannian manifold $M^n$ then $ξ$ naturally defines a submanifold $ξ(M^n)$ transverse to the fibers of the tangent bundle $TM^n$ with Sasaki metric. In this paper, we are interested in transverse totally geodesic submanifolds of the tangent bundle. We show that a transverse submanifold $N^l$ of $TM^n$ ($1 \leq l \leq n$) can be realized locally as the image of a submanifold $F^l$ of $M^n$ under some vector field $ξ$ defined along $F^l$. For such images $ξ(F^l)$, the conditions to be totally geodesic are presented. We show that t…
Totally geodesic Lagrangian submanifolds of the pseudo-nearly Kähler $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$
本文研究伪近凯勒流形 $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ 中的拉格朗日子流形，依据其关于近乘积结构的行为将其分为四类，并给出全测地拉格朗日子流形的完整分类。
An isometric immersion of a Riemannian manifold M into a Riemannian manifold N gives rise in a natural way to the immersion of the tangent bundle TM into the tangent bundle TN with a non-degenerate g- natural metric G.
Normal-Yang-Mills and Tangent-Yang-Mills submanifolds
本文研究子流形浸入单位球体时正法和切法曲率张量的L²范数变分问题，定义了正法-杨-米尔斯和切法-杨-米尔斯子流形，并推导了相应的欧拉-拉格朗日方程，构造了非平凡例子。
Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices
本文阐述了对称正定实矩阵流形 SPD 的几何，给出了子流形在仿射不变黎曼度量下为全测地的充要条件，定义了具有极小化性质的非线性投影，并导出了流形分解及在协方差矩阵分解中的应用。
It is proved, that if M is a connected, complete submanifold of a complex space form N and each geodesic of M lies in an 1-dimensional totally geodesic complex submanifold of N, then M is totally geodesic in N and is a real space form or a complex space form.
Totally geodesic submanifolds of a complex flag
该研究对配备标准齐性黎曼度量的复流形M = SU(n+2)/S(U(1)×U(1)×U(n))的极大全测地子流形进行分类，通过自然纤维化与曲率计算完成分类，得到含多种类型子流形的完整描述。
In this paper, we define and study CR-lightlike submanifolds of a golden semi-Riemannian manifold. We investigate some properties of geodesic CR-submanifolds of a golden semi-Riemannian manifold. Moreover, we obtain many interesting results for totally geodesic and totally umbilical CR-submanifolds on a golden Riemannian manifold. A non-trivial example of CR-lightlike submanifold of a golden semi-Riemannian manifold is also constructed.
In this paper we initiate the study of submanifolds of almost hypercomplex manifolds with Hermitian and Norden metrics. Object of investigations are holomorphic submanifolds of the hypercomplex manifolds which are locally conformally equivalent to the hyper-Kähler manifolds of the considered type. Necessary and sufficient conditions the investigated holomorphic submanifolds to be totally umbilical or totally geodesic are obtained. Examples of the examined submanifolds are constructed.
In the present note, first we derive an intrinsic inequality for Pseudo-umbilical spacelike submanifold in an indefinite space form. We use this inequality to show that such submanifold is totally geodesic. In the rest part of this paper, using a result of Aiyama [1], we prove that Pseudo-umbilical spacelike subamnifold is totally umbilical.
The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my previous papers cited in the present paper as [K1] and [K2], and for the space SU(3)/SO(3) in [K3], Section 6, we now solve the classification for the remaining irreducible Riemannian symmetric spaces of rank 2 and compact type: SU(6)/Sp(3), SO(10)/U(5), E6/(U(1)*Spin(10)), E6/F4, G2/SO(4), SU(3), Sp(2) and G2. Similarly as for the spaces already investigated in the…
A perspective on totally geodesic submanifolds of the symmetric space $G_2/SO(4)$
本文独立证明了对称空间 $G_2$ 和 $G_2/SO(4)$ 的极大全测地子流形分类，并通过主子代数、稳定化子、结合子代数及格拉斯曼流形等给出自然描述。
Random 3-Manifolds Have No Totally Geodesic Submanifolds
本文证明在紧致三维流形上，不具有全测地子流形的黎曼度量集合包含一个在C^q拓扑（q≥3）中开且稠密的子集，从而加强了Lytchak和Petrunin关于三维情形通用集仅为稠密G_δ集的结果。
In an earlier paper we discussed soldered forms, multivector fields and Riemannian metrics. In particular, we showed that a Riemannian submanifold is totally geodesic iff the metric is soldered to the submanifold. In the present note we discuss general, soldered tensor fields. In particular, we prove that the almost complex structure of an almost Kähler manifold is soldered to a submanifold iff the latter is an invariant, totally geodesic submanifold.
One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds $M_+,M_-$ introduced and studied by Chen and Nagano in [Totally geodesic submanifolds of symmetric spaces, II, Duke Math. J. 45 (1978), 405--425] as minimal submanifolds. The other purpose is to establish a stability theorem for minimal totally real submanifolds of Kählerian manifolds.
We derive total mean curvature integration formulae of a three co-dimensional foliation $\mathcal{F}^{n}$ on a screen integrable half-lightlike submanifold, $M^{n+1}$ in a semi-Riemannian manifold $\overline{M}^{n+3}$. We give generalized differential equations relating to mean curvatures of a totally umbilical half-lightlike submanifold admitting a totally umbilical screen distribution, and show that they are generalizations of those given by [4].
We prove that pseudo-holomorphic discs attached to a maximal totally real submanifold inherit their regularity from the regularity of the submanifold and of the almost complex structure. The proof is based on the computation of an explicit lower bound for the Kobayashi metric in almost complex manifolds, which also yields explicit estimates of Hölderian norms of such discs.
We introduce a new class of lightlike submanifolds, namely, Screen Transversal Cauchy Riemann (STCR)-lightlike submanifolds, of indefinite Kaehler manifolds. We show that this new class is an umbrella of screen transversal lightlike, screen transversal totally real lightlike and CR-lightlike submanifolds. We give a few examples of a STCR lightlike submanifold, investigate the integrability of various distributions, obtain a characterization of such lightlike submanifolds in a complex space form and find new conditions for the induced connection to be a metric connection. Moreover, we investiga…
We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kähler case, by its analogues. To this end we explore the geometry of totally real submanifolds, defining (i) a new geometric flow in terms of the ambient canonical bundle, (ii) a modified volume functional which takes into account the totally real condition. We discuss short-time existence for our flow and show it couples …
Doubly Totally-Umbilical Statistical Submanifolds in the Probability Simplex
本文在概率单纯形中完全分类了双全脐统计子流形，这是信息几何中统计子流形理论的一个重要问题。
Nonproduct Minimal Submanifolds with Cylindrical Tangent Cones
本文通过扰动方法构造无穷维族具有柱形切锥奇异集的非积极小子流形，解答了Leon Simon的问题，推广了Caffarelli-Hardt-Simon的工作，需用tame Fréchet空间范畴与Nash-Moser隐函数定理完成研究。
We write down the local equations that characterize the submanifolds N of a Dirac manifold M which have a normal bundle that is either a coisotropic or an isotropic submanifold of TM endowed with the tangent Dirac structure. In the Poisson case, these formulas prove again a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of TM with the tangent Poisson structure iff N is a Dirac submanifold. In the presymplectic case, it is the isotropy of the normal bundle which characterizes the corresponding notion of a Dirac submanifold. On the way, we give a simple de…
Totally Geodesic Submanifolds in Products of Non-Positively Curved Manifolds
研究非正曲率闭 manifold M 和 M×M 中满足正交条件的 n 维完全测地子流形，证明在 M 满足一定不可约性条件下，若 M×M 存在无限多个此类子流形或单个致密子流形，则 M 是局部对称空间。
This is the content of a talk given by the author at the 2009 Lehigh University Geometry/Topology Conference. Using the definition of connection given by Dieudonné, the Sasaki metric on the tangent bundle to a Riemannian manifold is expressed in a natural way. Also, the following property is established. The induced metric on the tangent bundle of an isometrically embedded submanifold is the Sasaki metric if and only if the submanifold is totally geodesic.
We investigate submanifolds in space forms such that every geodesic orthogonal to the submanifold intersects a fixed totally geodesic submanifold. We obtain an application to horospheres in Hadamard manifolds.
We present a new equation with respect to a unit vector field on Riemannian manifold $M^n$ such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and prove that within this class the Hopf vector field is a unique global one with totally geodesic property. For the wider class of geodesic unit vector fields on a sphere we give a new necessary and sufficient condition to generate a totally geodesic submanifold in $T_1S^n$.
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