Regularity of the geodesic flow on submanifolds
Christian Lange
本文证明了$\mathbb{R}^n$中$C^k$子流形的测地流和指数映射具有$C^{k-1}$类正则性。
我们证明，对于$\mathbb{R}^n$中$k\geq 2$的$C^k$子流形，其测地流和指数映射属于$C^{k-1}$类。
We show that the geodesic flow and the exponential map of a $C^k$ submanifold of $\mathbb{R}^n$ with $k\geq 2$ are of class $C^{k-1}$.
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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.
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 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.
In the present paper we study pseudo-Riemannian submanifolds which have 3-planar geodesic normal sections.We consider W-curves (helices) on pseudo-Riemannian submanifolds. Finally, we give neccessary and sufficient condition for a normal section to be a W-curve on pseudo-Riemannian submanifolds.
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…
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.
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…
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 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 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.
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。
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.
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…
A conformal structure on a manifold $M^n$ induces natural second order conformally invariant operators, called Möbius and Laplace structures, acting on specific weight bundles of $M$, provided that $n\ge 3$. By extending the notions of Möbius and Laplace structures to the case of surfaces and curves, we develop here the theory of extrinsic conformal geometry for submanifolds, find tensorial invariants of a conformal embedding, and use these invariants to characterize various forms of geodesic submanifolds.
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…
In this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime. The proof is based on both variational and geometric arguments involving the causal structure of the spacetime, the completeness of suitable Finsler metrics associated to it and some basic properties of a submersion. By this interaction, unlike previous results on the topic, also non--spacelike submanifolds can be handled.
In this note, we prove that given a submanifold $P$ in a Finsler manifold $(M,F)$, (i) the orthogonal geodesics to $P$ minimize the distance from $P$ at least in some interval, (ii) there exist tubular neighbourhoods around each point of $P$, (iii) the distance from $P$ is smooth in some open neighbourhood of $P$ (but not necessarily in $P$).
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.
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.
Geodesic normal coordinates and natural tensors for pseudo-Riemannian submanifolds
本文构造了适用于伪黎曼子流形的测地法坐标，证明度量泰勒系数可由背景曲率张量与第二基本形式的协变导数的通用多项式表示，并据此定义和刻画了自然子流形张量与微分算子。
The $L^2$-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type $M$ in a Riemannian manifold $(N,g)$ induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the $L^2$-metric.
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。
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 …
Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces
研究模2同调中面积最小化子流形的光滑性，证明其一般非光滑（除特定情况），解决怀特猜想，还建立奇异集豪斯多夫维数下界，并证明$\rpt$维罗内塞极小嵌入上的锥是模2面积最小化的。
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。
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…
We prove that there exists a metric of positive curvature in a three-sphere which admits a given torus knot as a closed geodesic.We also sketch a construction of a metric in a four sphere, very likely of positive curvature, which admits a totally geodesic projective plane with Euler number four. Surpisingly, the technique borrows a lot from the Mostow-Siu-Gromov-Thurston constuction of exotic negatively curved manifolds.
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。
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。
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.
$H$-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.
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.
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 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.
Properness and finiteness of totally geodesic submanifolds in the convex core
本文研究几何有限秩一局部对称流形凸核内的全测地子流形，证明其凸核内维数≥2的极大全测地子流形正常浸入、体积有限且数量有限，结合已有工作推导得出含无穷多该类子流形的对应流形必为算术流形。
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。
Simplicial volume and isolated, closed totally geodesic submanifolds of codimension one
本文证明含余维数一孤立闭全测地子流形的非正曲率闭流形具有正单纯体积，并推出非正弯曲解析流形在万有覆叠含余维数一平坦时的二分结论。
On the Cut Locus of Submanifolds of a Finsler Manifold
本文研究前向完备Finsler流形中闭子流形的割迹，通过引入N-测地环概念，将Klingenberg闭测地线引理推广到可逆Finsler情形下的N-测地环。
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。
We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally geodesic subvarieties. As a corollary, we obtain that on a complex surface which is a compact quotient of the bidisc or of the 2-ball, there is at most a finite number of totally geodesic curves with negative self intersection. More generally, we prove that there are only finitely many exceptional totally geodesic divisors in a compact Hermitian locally symmetric …
We discover new monotonicity formulae for minimal submanifolds in space forms, which imply the sharp area bound for minimal submanifolds through a prescribed point in a geodesic ball. These monotonicity formulae involve an energy-like integral over sets which are, in general, not geodesic balls. In the Euclidean case, these sets reduce to the moving-centre balls introduced by the second author in [Zhu18].
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。
In hyperbolic space $H^n$ we set a geodesic ball of radius $ρ$. Consider a $k$ dimensional minimal submanifold passing through the origin of the geodesic ball with boundary lies on the boundary of that geodesic ball. We prove that its area is no less than the totally geodesic $k$ dimensional submanifold passing through the origin in that geodesic ball. This is a partial generalization of the corresponding problem in $R^n$.
Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to achieve a strong "Colding-Minicozzi" type min-max approximation result.
The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse's proof. This approach reduces substantially the effort required in the proofs of the Theorem given in previous articles on the subject. Exactly the same argument works also in the case of timelike geodesics between two submanifolds of a Loren…
Efficient approximation of geodesics is crucial for practical algorithms on manifolds. Here we introduce a class of retractions on submanifolds, induced by a foliation of the ambient manifold. They match the projective retraction to the third order and thus match the exponential map to the second order. In particular, we show that Newton retraction (NR) is always stabler than the popular approach known as oblique projection or orthographic retraction: per Kantorovich-type convergence theorems, the superlinear convergence regions of NR include those of the latter. We also show that NR always ha…
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。
Let M be a possibly non compact smooth manifold. We study genericity in the C^k-topology (3<=k<=+infty) of nondegeneracy properties of semi-Riemannian geodesic flows on M. Namely, we prove a new version of the Bumpy Metric Theorem for a such M and also genericity of metrics that do not possess any degenerate geodesics satisfying suitable endpoints conditions. This extends results in the literature for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P of MxM that satisfies an admissibility condition. Immediate consequences are generic non conjugacy betwee…
We prove that the Hopf vector field is a unique one among geodesic covariantly normal unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation using standard approach from the theory of submanifolds and find exact boundaries for the sectional curvature of the Hopf vector field.
On free boundary minimal submanifolds in geodesic balls in $\mathbb H^n$ and $\mathbb S^n_+$
本文在双曲空间和上半球面的测地球中研究自由边界极小子流形，定义了两个自然泛函并证明其临界度量与自由边界极小浸入诱导度量之间的对应关系，同时计算了若干关键例子的 Morse 指标。
Let H be the hyperbolic space of dimension n+1. A geodesic foliation of H is given by a smooth unit vector field on H all of whose integral curves are geodesics. Each geodesic foliation of H determines an n-dimensional submanifold M of the 2n-dimensional manifold L of all the oriented geodesics of H (up to orientation preserving reparametrizations). The space L has a canonical split semi-Riemannian metric induced by the Killing form of the isometry group of H. Using a split special Lagrangian calibration, we study the volume maximization problem for a certain class of geometrically distinguish…
In this article, we construct a new para-Kähler structure $({\mathcal G},{\mathcal J},Ω)$ in the space of oriented geodesics ${\mathbb L}(M)$ in a non-flat, real space form $M$. We first show that the para-Kähler metric ${\mathcal G}$ is scalar flat and when $M$ is a 3-dimensional real space form, ${\mathcal G}$ is locally conformally flat. Furthermore, we prove that the space of oriented geodesics in hyperbolic $n$-space, equipped with the constructed metric ${\mathcal G}$, is minimally isometric embedded in the tangent bundle of the hyperbolic $n$-space. We then study the submanifold theory,…
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 $T_1S^n$.
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.
Planar pseudo-geodesics and totally umbilic submanifolds
本文研究 Riemannian 流形间的全脐等距浸入，通过引入平面伪测地线这一自然外蕴推广，给出了具有平行单位平均曲率向量的全脐等距浸入的新刻画，并证明了余维为一时浸入全脐当且仅当所有测地线的外蕴形状为平面，推广了三维欧氏空间曲面的经典结果。
In this article the framework created by Cartan to produce local differential invariants for submanifolds of homogeneous spaces is applied to classify all totally geodesic Lagrangian submanifolds and all homogeneous Lagrangian submanifolds of the nearly Kähler manifold of full flags in $\mathbb{C}^3$.
We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded minimal submanifolds in simply connected noncompact globally symmetric spaces.
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