Mirjana Milijevic
Let M be a CR submanifold of maximal CR dimension of a complex space form M. The shape operator A of the distinguished vector field ξ is recurrent if there exists a 1-form v such that \nabla A = A \otimes v. We show that M is an Euclidean space under the condition that A is recurrent.
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The properties of Kaehler submanifolds with recurrent the second fundamental form in spaces of constant holomorphic sectional curvature are being studied in this article.
The complete local classification and geometric description of n-dimensional submanifolds F with recurrent nonparallel second fundamental form in the spaces of constant curvature M(c) are obtained in this article.
Motivated by spectral asymptotics for orbital integrals in a relative trace formula, we generalize a number of geometric properties of geodesics in the hyperbolic plane, to maximal flat submanifolds of symmetric spaces of non-compact type.
On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.
In this paper, we study the CR submanifolds of maximal CR dimension with flat normal connection of a complex projective space. We first investigate the position of the umbilical normal vector in the normal bundle, especially for the submanifolds of dimension 3. Then as the application, we prove the non-existence of a class of CR submanifolds of maximal CR dimension with flat normal connection.
Austere submanifolds in Euclidean space were introduced by Harvey and Lawson in connection with their study of calibrated geometries. The algebraic possibilities for second fundamental forms of 4-dimensional austere submanifolds were classified by Bryant, into three types which we label A, B, and C. In this paper, we show that type A submanifolds correspond exactly to real Kahler submanifolds, we construct new examples of such submanifolds in R^6 and R^10, and we obtain classification results on submanifolds of types B and C with maximal second fundamental forms.
It has been proved that there are no real hypersurfaces satisfying RA = 0 in non-flat complex space forms. In this paper we prove that the same is true in the case of CR submanifolds of maximal CR dimension, that is there are no CR submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex space forms.
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 ${\rm U}(n)$.
On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces
本文研究了伪双曲空间中完备极大类空子流形的曲率，证明了其标量曲率在任意符号下均非正并给出精确界限，显式刻画了达到界限的子流形，并在余维数为1时推导出Ricci曲率的精确上界。
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 …
We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of the canonical distribution ("lagrangians") and illustrate some notable ones of small dimension. An infinitesimal classification of the arbitrary maximal horizontal submanifolds follows as a consequence.
We classify pseudo parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one. With this result, the non-existence of recurrent as well as semi parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one can also be obtained.
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 note we consider the Liouville type theorem for a properly immersed submanifold $M$ in a complete Riemmanian manifold $N$. Assume that the sectional curvature $K^N$ of $N$ satisfies $K^N\geq-L(1+dist_N(\cdot,q_0)^2)^\fracα{2}$ for some $L>0, 2>α\geq 0$ and $q_0\in N$. (i) If $Δ|\vec{H}|^{2p-2}\geq k|\vec{H}|^{2p}$($p>1$) for some constant $k>0$, then we prove that $M$ is minimal. (ii) Let $u$ be a smooth nonnegative function on $M$ satisfying $Δu\geq ku^a$ for some constant $k>0$ and $a>1$. If $|\vec{H}|\leq C(1+dist_N(\cdot,q_0)^2)^\fracβ{2}$ for some $C>0$, $0\leqβ<1$, then $u=0$ on …
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。
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.
On complete maximal submanifolds in pseudo-hyperbolic space
本文完整分类了伪双曲空间中的完备极大类空子流形，并探讨了其在Teichmüller理论与双曲群Anosov表示理论中的应用。
We prove an isoperimetric-type inequality for maximal, spacelike submanifold in the Minkowski space. The argument is based on the recent work of Brendle.
Restriction of eigenfunctions on products of spheres to submanifolds of maximal flats
该研究确立了在紧致类型秩一对称空间积上，特征函数限制到最大平坦子流形的L^p界，并结合杰科比多项式渐进行为和球面函数傅里叶系数正性进行证明。
In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under certain condition, and derive the constancy of the principal curvatures of such submanifolds. As its result, we can derive that such submanifolds are isoparametric.
Some results on calibrated submanifolds in Euclidean space of cohomogeneity one and two
本文研究了在欧几里得空间中具有共效性的子流形，通过构造特殊Lagrangian子流形并证明其刚性结果。
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.
Extrinsically Symmetric Spaces, Submanifolds of Clifford Type and a Theorem of Harish-Chandra
本文证明欧氏空间中紧致内在对称子流形为外在对称当且仅当其极大环面为Clifford环面，并以此给出Harish-Chandra定理的几何证明。
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。
Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids
该研究证明大类非算术双曲n-流形（含Gromov等的混合流形）的有限体积浸入全测地超曲面有限，且高余维下维数≥2的极大测地子流形有限，证明结合算术群结构、动力系统与负曲率几何。
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。
Rigidity Results for Compact Submanifolds with Pinched Ricci Curvature in Euclidean and Spherical Space Forms
本文研究了在欧几里得和球面空间形式中具有受限曲率的紧致子流形，证明其要么是爱因斯坦克利福德 тор斯，要么是拓扑球体，或具有部分同调群消失，扩展了多项经典刚性结果。
We consider the Dirichlet boundary value problem for graphical maximal submanifolds inside Lorentzian type ambient spaces, and obtain general existence and uniqueness results which apply to any codimension.
We prove that there does not exist global-in-time axisymmetric solutions to the time-like minimal submanifold system in Minkowski space. We further analyze the limiting geometry as the maximal time of existence is approached.
We study the possible singularities of an $m$-subharmonic function $φ$ along a complex submanifold $V$ of a compact Kähler manifold, finding a maximal rate of growth for $φ$ which depends only on $m$ and $k$, the codimension of $V$. When $k < m$, we show that $φ$ has at worst log poles along $V$, and that the strength of these poles is moveover constant along $V$. This can be thought of as an analogue of Siu's theorem.
We give a conceptual proof of the fact that if M is a complete submanifold of a space form, then the maximal integral manifolds of the nullity distribution of its second fundamental form through points of minimal index of nullity are complete.
Motivated by Gray's work on tube formulae for complex submanifolds of complex projective space equipped with the Fubini-Study metric, Riemannian foliations of projective space are studied. We prove that there are no complex Riemannian foliations of any open subset of $\mathbb{P}^n$ of codimension one. As a consequence there is no Riemannian foliation of the projective plane by Riemann surfaces, even locally. We determine how a complex submanifold may arise as an exceptional leaf of a non-trivial singular Riemannian foliation of maximal dimension. Gray's tube formula is applied to obtain a volu…
We discuss a recurrent geometrical method, due to Élie Cartan and von Weber ([1],[11]) enabling us to determine, step by step, the maximal integral manifolds of a not necessarily integrable nor regular Pfaffian system. The dimensions of such integral manifolds can, of course, vary from point to point but more so can vary at a given point it depending upon the choice of their recurrent buildup. When the system is regular and integrable then, of course, we obtain the maximal integral leaves of the integral foliation. Attention is also given to those integrable systems that can be integrated by q…
Let $Γ$ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $Γ$ is arithmetic. This answers a question of Reid for hyperbolic $n$-manifolds and, independently, McMullen for hyperbolic $3$-manifolds. We prove these results by proving a superrigidity theorem for certain representations of such lattices. The proof of our superrigidity theorem uses results on equidistribution from homogeneous dynamics and our main result also admits a formulation in that language.
In this article, we solve the equivalence problem for 2--nondegenerate CR geometries that have (at every point) a homogeneous space $G/H$ as a maximally symmetric model for $G$ simple real Lie group of CR automorphisms. This completes the classification of real submanifolds in complex space that are maximally symmetric models with a real simple CR automorphism group. In particular, we construct (local) embeddings of these models into complex space.
Maximal Normal Curvature and Veronese Rigidity
证明闭浸入子流形在内在调和结构假设下的尖锐Veronese刚性定理，通过Bochner-Gauss机制将调和形式的Bochner曲率项转化为形状算子的尖锐代数估计。
It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let M^n, n>=2, be a full and irreducible homogeneous submanifold of the sphere $S^{N-1}\subset R^N$ and such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then M^n must be an orbit of an irreducible s-representation (i.e. the isotropy representation of a semisimple Riemannian symmetric space). If n=2, then the normal holonomy is always transitive, unless M is a homogeneous isoparam…
Let M be a compact irreducible Hermitian symmetric space and write M=G/K, with G the group of holomorphic isometries of M and K the stability group of the point of 0 in M. We determine the maximal dimension of a complex projective space embedded in M as a totally geodesic submanifold.
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkahler and G_2-manifolds respectively. For vector bundles over such manifolds, we introduce (special) A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Simila…
On the inverse mean curvature flow by parallel hypersurfaces in space forms
研究在空间形式中由平行超曲面进行反平均曲率流的存在条件，揭示了等参数超曲面的流解特性及塌缩行为。
Moduli spaces of holomorphic disks in a complex manifold Z, with boundaries constrained to lie in a maximal totally real submanifold P, have recently been found to underlie a number of geometrically rich twistor correspondences. The purpose of this paper is to develop a general Fredholm regularity criterion for holomorphic curves-with-boundary, and then show how this applies, in particular, to various moduli problems of twistor-theoretic interest.
A class of codimension one foliations has been recently introduced by imposing a natural compatibility condition with a closed maximally non-degenerate 2-form. In this paper we study for such foliations the information captured by a Donaldson type submanifold. In particular we deduce that their leaf spaces are homeomorphic to leaf spaces of 3-dimensional taut foliations. We also introduce surgery constructions to show that this class of foliations is broad enough. Our techniques come mainly from symplectic geometry.
We give a detailed account of the geometric correspondence between a smooth complex projective quadric hypersurface $\mathcal{Q}^n$ of dimension $n \geq 3$, and its twistor space $\mathbb{PT}$, defined to be the space of all linear subspaces of maximal dimension of $\mathcal{Q}^n$. Viewing complex Euclidean space $\mathbb{CE}^n$ as a dense open subset of $\mathcal{Q}^n$, we show how local foliations tangent to certain integrable holomorphic totally null distributions of maximal rank on $\mathbb{CE}^n$ can be constructed in terms of complex submanifolds of $\mathbb{PT}$. The construction is ill…
Maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$
研究Weil-Petersson类在3维反de Sitter空间中的最大圆盘，证明Mess映射在Weil-Petersson万有Teichmüller空间的余切 bundle 上的辛微分同胚，并展示反共轭能量函数作为Kähler势的作用。
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