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We find a one-to-one correspondence between full extrinsic symmetric spaces in (possibly degenerate) inner product spaces and certain algebraic objects called (weak) extrinsic symmetric triples. In particular, this yields a description of arbitrary extrinsic symmetric spaces in pseudo-Euclidean spaces by corresponding infinitesimal objects.
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Improving a result of Eschenburg and Kim we give a criterion for semisimplicity of pseudo-Riemannian extrinsic symmetric spaces in terms of the shape operator with respect to the mean curvature vector.
Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the classification difficult. We use the recently developed method of quadratic extensions for (h,K)-invariant metric Lie algebras to tackle this problem. We obtain a one-to-one correspondence between isometry classes of extrinsic symmetric spaces and a certain cohomology set. This all…
We determine all indecomposable pseudo-Riemannian symmetric spaces of signature (4,3) whose holonomy is contained in $G_{2(2)}\subset SO(4,3)$.
We classify the connected pseudo-Riemannian manifolds of signature $(p,q)$ with $q\ge5$ so that at each point of $M$ the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least one point of $M$.
The geometric characteristics of SGL submanifolds in an indefinite Sasakian statistical manifold equipped with a quarter symmetric metric connection
本文研究了在特定几何结构下SGL子流形的几何特性，分析了其分布的可积性和平行性，并通过示例展示了其应用。
We classify indefinite simply connected hyper-Kaehler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m,4m). We establish a natural 1-1 correspondence between simply connected hyper-Kaehler symmetric spaces of dimension 8m and orbits of the general linear group GL(m,H) over the quaternions on the space (S^4C^n)^τ of homogeneous quartic polynomials S in n = 2m complex variables satisfying the reality condition S = τS, where τis the real structure induced by the quaternionic structure of C^{2m} = H^m. We define and classify also complex hyper-K…
On the curvature symmetries in lightlike hypersurfaces of an indefinite Sasakian manifold
本文研究不定 Sasakian 流形中相切于结构向量场的类光超曲面，证明其不具备局部对称、半对称或半平行等曲率对称性。
This paper is a survey of results obtained by the authors on the geometry of connections with totally skew-symmetric torsion on the following manifolds: almost complex manifolds with Norden metric, almost contact manifolds with B-metric and almost hypercomplex manifolds with Hermitian and anti-Hermitian metric.
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。
Let $M$ be a symplectic symmetric space, and let $\imath : M \to V$ be an extrinsic symplectic symmetric immersion, i.e., $(V, Ω)$ is a symplectic vector space and $\imath$ is an injective symplectic immersion such that for each point $p \in M$, the geodesic symmetry in $p$ is compatible with the reflection in the affine normal space at $\imath(p)$. We show that the existence of such an immersion implies that the transvection group of $M$ is solvable.
Analogous to the bowl soliton of mean curvature flow, we construct rotationally symmetric translating solutions to a very large class of extrinsic curvature flows, namely those whose speeds are $α$-homogeneous ($α>0$), elliptic and symmetric with respect to the principal curvatures. We show that these solutions are necessarily convex, and give precise criteria for the speed functions which determine whether these translators are defined on all of $\mathbb{R}^{n}$ or contained in a cylinder. For speeds that are nonzero when at least one of the principal curvatures is nonzero, we are also able t…
Following the approach to pseudo-Riemannian symmetric spaces developed in math.DG/0408249 we exhibit examples of indefinite hyper-Kaehler symmetric spaces with non-abelian holonomy. Moreover, we classify indecomposable hyper-Kaehler symmetric spaces whose metric has signature (4,4n). Such spaces exist if and only if n=0,1,3.
We study infinitesimal semi-simple extrinsic symmetric spaces and give a classification in the symplectic case.
We classify those curvature-homogeneous Einstein four-manifolds, of all metric signatures, which have a complex-diagonalizable curvature operator. They all turn out to be locally homogeneous. More precisely, any such manifold must be either locally symmetric or locally isometric to a suitable Lie group with a left-invariant metric. To show this we explicitly determine the possible local-isometry types of manifolds that have the properties named above, but are not locally symmetric.
Due to its length and several inaccuracies, this article is no longer suggested for reading. Moreover, in the meantime certain results presented herein could be improved by the author in a non-trivial fashion. Instead, the reader is referred to the following two articles: The extrinsic holonomy Lie algebra of a parallel submanifold (arXiv:0904.2611), Extrinsic homogeneity of parallel submanifolds (arXiv:0904.2636).
We consider parallel submanifolds $M$ of a Riemannian symmetric space $N$ and study the question whether $M$ is extrinsically homogeneous in $N$\,, i.e.\ whether there exists a subgroup of the isometry group of $N$ which acts transitively on $M$\,. First, given a "2-jet" $(W,b)$ at some point $p\in N$ (i.e. $W\subset T_pN$ is a linear space and $b:W\times W\to W^\bot$ is a symmetric bilinear form)\,, we derive necessary and sufficient conditions for the existence of a parallel submanifold with extrinsically homogeneous tangent holonomy bundle which passes through $p$ and whose 2-jet at $p$ is …
We investigate parallel submanifolds of a Riemannian symmetric space $N$. The special case of a symmetric submanifold has been investigated by many authors before and is well understood. We observe that there is an intrinsic property of the second fundamental form which distinguishes full symmetric submanifolds from arbitrary full parallel submanifolds of $N$, usually called "1-fullness of $M$". Furthermore, for every parallel submanifold $M$ of $N$ we consider the pullback bundle $TN|M$ with its induced connection, which admits a distinguished parallel subbundle $osc M$, usually called the "s…
Let $(V, \Om)$ be a symplectic vector space and let $ϕ: M \ra V$ be a symplectic immersion. We show that $ϕ(M) \subset V$ is (locally) an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of \cite{CGRS} if and only if the second fundamental form of $ϕ$ is parallel. Furthermore, we show that any symmetric space which admits an immersion as an e.s.s.s. also admits a {\em full} such immersion, i.e., such that $ϕ(M)$ is not contained in a proper affine subspace of $V$, and this immersion is unique up to affine equivalence. Moreover, we show that any extrinsic symplectic immersion of $M$…
Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form $\langle\cdot,\cdot\rangle$. We assume that $\langle\cdot,\cdot\rangle $ is nil-invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint representation of $\mathfrak{g}$ is an infinitesimal isometry for $\langle\cdot,\cdot\rangle $. Among these Lie algebras are the isometry Lie algebras of pseudo-Riemannian manifolds of finite volume. We prove a strong invariance property for nil-invariant symmetric bilinear forms, whic…
Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications
针对完全非线性外蕴曲率流的平移孤子，发展旋转理论，获得碗型平移子的渐近展开，并引入符号颈框架构造和分类悬链面型平移子，进而证明严格凸整体图平移子的唯一性和有界图平移子的悬链面障碍不存在性。
Let $M$ be a simply connected pseudo-Riemannian homogeneous space of finite volume with isometry group $G$. We show that $M$ is compact and that the solvable radical of $G$ is abelian and the Levi factor is a compact semisimple Lie group acting transitively on $M$. For metric index less than three, we find that the isometry group of $M$ is compact itself. Examples demonstrate that $G$ is not necessarily compact for higher indices. To prepare these results, we study Lie algebras with abelian solvable radical and a nil-invariant symmetric bilinear form. For these, we derive an orthogonal decompo…
Almost symmetric submanifolds
本文研究了欧几里得空间中的近对称子流形，揭示了其与对称子流形和子黎曼流形的关系，并提出了新的不变量用于研究高对称性子流形。
In this paper we introduce the concept of $(\varepsilon)$-almost paracontact manifolds, and in particular, of $(\varepsilon)$-para Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of $(\varepsilon)$-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it can not admit an $(\varepsilon)$-para Sasakian structure. We show that, for an $(\varepsilon)$-para Sasakian manifold, the conditions of being symmetric, semi-symmetric or of constant se…
Equivariant Embeddings of Kälerian Symmetric Spaces
本文研究对称凯勒流形的等变嵌入性质，在$\mathbb{CP}^n$情形下刻画了这些嵌入，并证明了具有平行多重均值曲率的等变嵌入必为外在对称的。
Symmetric Spaces with Rectangular Unit Lattices, Revisited
本文通过显式构造等距外蕴对称嵌入，重新证明了Loos关于矩形单位格黎曼对称空间为对称R-空间的定理，并直接确定了其根系、基本群及拉普拉斯特征值。
The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers with interest in applications. Secondly, we concentrate on rolling an important class of Riemannian manifolds. In the first part of the paper, the relation between intrinsic and extrinsic rollings is explained in detail, while in the second part we address rollings of symmetric spaces on flat spaces and complement the theoretical results with illustrative exam…
In this paper we provide a method to study critical points of strongly indefinite functionals on vector bundles. We focus mainly on energy functionals coupled with a fermionic part, that is with a Dirac-type operator. We consider the cases of the perturbed Dirac-Geodesics problem and the Yang-Mills-Dirac type equation in dimension two.
Given a complete isometric immersion $ϕ: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space $M^n_w$, we determine a set of conditions on the extrinsic curvatures of $P$ that guarantees that the immersion is proper and that $P$ has finite topology, in the line of the paper "On Submanifolds With Tamed Second Fundamental Form", (Glasgow Mathematical Journal, 51, 2009), authored by G. Pacelli Bessa and M. Silvana Costa. When the ambient ma…
In a noncompact harmonic manifold $M$ we establish finite dimensionality of the eigenspaces $V_λ$ generated by radial eigenfunctions of the form $\cosh r + c$. As a consequence, for such harmonic manifolds, we give an isometric imbedding of $M$ into $(V_λ,B)$, where $B$ is a nondegenerate symmetric bilinear indefinite form on $V_λ$ (analogous to the imbedding of the real hyperbolic space $I\!\!\!H^{n}$ into $I\!\!\!R^{n+1}$ with the indefinite form $Q(x,x) = -x_0^2 + \sum x_i^2$). Finally we give certain conditions under which $M$ is symmetric.
In this paper we introduce a flow on the spectral data for symmetric CMC surfaces in the $3$-sphere. The flow is designed in such a way that it changes the topology but fixes the intrinsic (metric) and certain extrinsic (periods) closing conditions of the CMC surfaces. For rational times we obtain closed (possibly branched) connected CMC surfaces of higher genus. We prove the short time existence of this flow near the spectral data of (a class of) CMC tori. In particular we prove that flowing the spectral data for the Clifford torus is equivalent to the flow of Plateau solutions by varying the…
Symmetric spaces for groups over involutive algebras and applications to Higgs bundles
研究对合代数上的辛群和不定正交群，识别其极大紧子群并构造黎曼对称空间的几何模型，进而应用于Higgs丛模空间的计数和Hitchin态射的分解。
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