本文通过将该定理视为高余维的Cartan-Ambrose-Hicks定理，将子流形的基本定理推广到一般环境空间。实现这一结果的关键在于正余维情形下曲线展开（development of curves）的推广。我们结果的一个优点是，当等距浸入存在时，它也提供了等距浸入的几何构造。
英文摘要：
In this paper, we extend the fundamental theorem for submanifolds to general ambient spaces by viewing it as a higher codimensional Cartan-Ambrose-Hicks theorem. The key ingredient in obtaining this is a generalization of development of curves in the positive codimensional case. One advantage of our results is that it also provide a geometric construction of the isometric immersion when the isometric immersion exists.
In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowski's generalized formula we can focus on several isoperimetric problems. More precisely, on lower bounds for isoperimetric quotients of any precompact domain with smooth boundary, or on the isoperimetric profile of the submanifold and its modified volume. In the particular case of a model space with st…
Given a Kaehler manifold of complex dimension 4, we consider submanifolds of (real) dimension 4, whose Kaehler angles coincide. We call these submanifolds Cayley. We investigate some of their basic properties, and prove that (a) if the ambient manifold is a Calabi-Yau, the minimal Cayley submanifolds are just the Cayley submanifolds as defined by Harvey and Lawson; (b) if the ambient is a Kaehler-Einstein manifold of non-zero scalar curvature, then minimal Cayley submanifolds have to be either complex or Lagrangian.
In this paper, we prove a Sobolev and isoperimetric inequalities for submanifold in weighted manifold. Our results generalize the Hoffman-Spruck's inequalities.
In this paper we study a geometric configuration of submanifolds of arbitrary codimension in an ambient Riemannian space. We obtain relations between the geometry of a q-codimension submanifold Mn along its boundary and the geometry of the boundary of Mn as an hypersuface of a q-codimensional submanifold Pn in an ambient space Mn+q. As a consequence of these geometric ralations we get that the ellipticity of the generalized Newton transformations implies the tranversality of Mn and Pn in Pn is totally geodesic in Mn+q.
We introduce a new technique to the study and identification of submanifolds of simply-connected symmetric spaces of compact type based upon an approach computing $k$-positive Ricci curvature of the ambient manifolds and using this information in order to determine how highly connected the embeddings are. This provides codimension ranges in which the Cartan type of submanifolds satisfying certain conditions which generalize being totally geodesic necessarily equals the one of the ambient manifold. Using results by Guijarro--Wilhelm our approach partly generalizes recent work by Berndt--Olmos o…
We develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. In particular, we establish the subtle relationship between the submanifold and ambient standard tractor bundles, allowing us to relate the respective normal Cartan (or tractor) connections via a CR Gauss formula. The CR invariant tractor calculus of CR embeddings is developed concretely in terms of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient contact form. This enables straightforward and explicit calculation o…
We classify homogeneous CR submanifolds in complex hyperbolic spaces arising as orbits of a subgroup of the solvable part of the Iwasawa decomposition of the isometry group of the ambient space.
We extend the estimate obtained in [1] for the mean curvature of a cylindrically bounded proper submanifold in a product manifold with an Euclidean space as one factor to a general product ambient space endowed with a warped product structure.
Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. The main purpose of this paper is to completely classify all non-minimal ideal submanifolds of real space forms with type number $\leq 2$.
In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not admit fillings by holomorphic discs. The proof relies on a mix of holomorphic curve techniques and on certain convexity results.
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 …
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…
We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean cu…
We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in \cite{dksv1} and \cite{dksv2}, respectively. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of $\Hy^n$.
It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-adapted isoparametric submanifold and its focal submanifold. Concretely the investigation is performed by investigating the mean curvature flows for the lift of the submanifold to an infinite dimensional pseudo-Hilbert space through a pseudo-Riemannian submersion.
An isometric immersion $f:M^n\to \tilde M^n$ from a Riemannian $n$-manifold $M^n$ into a Kähler $n$-manifold $\tilde M^n$ is called {\it Lagrangian} if the complex structure $J$ of the ambient manifold $\tilde M^n$ interchanges each tangent space of $M^n$ with the corresponding normal space. In this paper, we completely classify spherical Lagrangian submanifolds in complex Euclidean spaces. Furthermore, we also provide two corresponding classification theorems for Lagrangian submanifolds in the complex pseudo-Euclidean spaces with arbitrary complex index.
Given a Riemannian manifold $N^n$ and ${\cal Z}\in \mathfrak{X}(N)$, an isometric immersion $f\colon M^m\to N^n$ is said to have the \emph{constant ratio property with respect to ${\cal Z}$} either if the tangent component ${\cal Z}^T_f$ of ${\cal Z}$ vanishes identically or if ${\cal Z}^T_f$ vanishes nowhere and the ratio $\|{\cal Z}^\perp_f\|/\|{\cal Z}^T_f\|$ between the lengths of the normal and tangent components of ${\cal Z}$ is constant along $M^m$. It has the \emph{principal direction property with respect to ${\cal Z}$} if ${\cal Z}^T_f$ is an eigenvector of all shape operators of $f$…
We prove an extension of a theorem of A.Ros on a characterization of seven compact Kaehler submanifolds by holomorphic pinching to certain submanifolds of the complex Grassmannian manifolds.
We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a C^1 dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a C^1 neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem [arXiv:1605.03645] which applies only to calibrated…
In this paper, we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups. In particular, we obtain the uniqueness and existence theorems, and some rigidity theorems.
The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundamental tone of minimal submanifolds in a Cartan-Hadamard ambient manifold. As an intrinsic result, we obtain a sufficient condition on the volume growth of a Cartan-Hadamard manifold to achieve the lowest bound for the fundamental tone given by McKean.
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 study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question.
We prove that every (compact) taut submanifold in Euclidean space is real algebraic, i.e., is a connected component of a real irreducible algebraic variety in the same ambient space. This answers affirmatively a question of Nicolaas Kuiper raised in the 1980s.
We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_{N}\leq κ\leq 0$ are proper, (compact if $N$ is compact). In addition, if $N$ is Hadamard then $M$ has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realized as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.
A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is a parallel section of the appropriate tensor bundle. We classify parallel submanifolds of the Grassmannian $\rmG^+_2(\R^{n+2})$ which parameterizes the oriented 2-planes of the Euclidean space $\R^{n+2}$\,. Our main result states that every complete parallel submanifold of $\rmG^+_2(\R^{n+2})$\,, which is not a curve, is contained in some totally geodesic submanifold as a symmetric submanifold. This result holds also if the ambient space is the non-compact dual of $\rmG^+_2(\R^{n+2})$\,.
We classify Lagrangian submanifolds of complex space forms, whose second fundamental form can be written in a certain way, depending on a real parameter. For some special values of this parameter, the resulting submanifolds are ideal in the sense that they realize equality in an inequality for a Chen's delta-curvature.
The space of embedded submanifolds plays an important role in applications such as computational anatomy and shape analysis. We can define two different classes on Riemannian metrics on this space: so-called outer metrics are metrics that measure shape changes using deformations of the ambient space and they find applications mostly in computational anatomy; the second class that are defined directly on the space of embeddings using intrinsic differential operations and they are used in shape analysis. In this paper we compare for the first time the topologies and the geodesic distance functio…
If M is a submanifold of a space form, the nullity distribution N of its second fundamental form is (when defined) the common kernel of its shape operators. In this paper we will give a local description of any submanifold of the Euclidean space by means of its nullity distribution. We will also show the following global result: if M is a complete, irreducible submanifold of the Euclidean space or the sphere then N is completely non integrable. This means that any two points in M can be joined by a curve everywhere perpendicular to N. We will finally show that this statement is false for a sub…
We prove an equivariant implicit function theorem for variational problems that are invariant under a varying symmetry group (corresponding to a bundle of Lie groups). Motivated by applications to families of geometric variational problems lacking regularity, several non-smooth extensions of the result are discussed. Among such applications is the submanifold problem of deforming the ambient metric preserving a given variational property of a prescribed family of submanifolds, e.g., constant mean curvature, up to the action of the corresponding ambient isometry groups.
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…
We prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then, we prove the existence of associated families of minimal surfaces in such products. Finally, in the case of $\mathbb{S}^2\times\mathbb{S}^2$, we give a complex version of the main theorem in terms of the two canonical complex structures of $\mathbb{S}^2\times\mathbb{S}^2$.
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.
Minimal Submanifolds of The Complex and Quaternionic Projective and Hyperbolic Spaces $\cn P^{2n-1}$, $\hn P^{2n-1}$, $\cn H^{2n-1}$, $\hn H^{2n-1}$ via Harmonic Morphisms
In this paper, we prove a classification theorem for the stable compact minimal submanifolds of the Riemannian product of an $m_1$-dimensional ($m_1\geq3$) hypersurface $M_1$ in the Euclidean space and any Riemannian manifold $M_2$, when the sectional curvature $K_{M_1}$ of $M_1$ satisfies $\frac{1}{\sqrt{m_1-1}}\leq K_{M_1}\leq 1.$ This gives a generalization to the results of F. Torralbo and F. Urbano [9], where they obtained a classification theorem for the stable minimal submanifolds of the Riemannian product of a sphere and any Riemannian manifold. In particular, when the ambient space is…
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.
We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive…
We give an optimal estimate for the norm of any submanifold's second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold. This is a special case of a similar theorem for intermediate Ricci curvature, and leads to a $C^{1,α}$ compactness result for submanifolds, as well as a "soul-type" structure theorem for manifolds with nonnegative $k^{th}$--intermediate Ricci curvature that have a closed submanifold with dimension $\geq k$ and infinite focal radius. To prove these results, we use the comparison lemma for Jacobi fields from arXiv:1603…
On Kahler 4-manifolds, not necessarily compact or of finite topological type, we obtain relationships between the fundamental group of compact embedded Levi-flat or pseudoconvex submanifold and the fundamental group of the ambient manifold $M^4$. When a Levi-flat submanifold $V^3$ has finite fundamental group then $π_1(M^4)=ι_*π_1(V^3)$; when a non-separating pseudoconvex submanifold $V^3$ has finite fundamental group, then $π_1(M^4)=ι_*π_1(V^3)\rtimes\mathbb{Z}$. As applications, if a Kahler manifold (compact or not) has an embedded holomorphic $\mathbb{P}^1$ of positive self-intersection, it…
Let $M$ be an $n$-dimensional Lagrangian submanifold of a complex space form. We prove a pointwise inequality $$δ(n_1,\ldots,n_k) \leq a(n,k,n_1,\ldots,n_k) \|H\|^2 + b(n,k,n_1,\ldots,n_k)c,$$ with on the left hand side any delta-invariant of the Riemannian manifold $M$ and on the right hand side a linear combination of the squared mean curvature of the immersion and the constant holomorphic sectional curvature of the ambient space. The coefficients on the right hand side are optimal in the sence that there exist non-minimal examples satisfying equality at at least one point. We also character…
We define and study isoparametric submanifolds of general ambient spaces and of arbitrary codimension. In particular we study their behaviour with respect to Riemannian submersions and their lift into a Hilbert space. These results are used to prove a Chevalley type restriction theorem which relates by restriction eigenfunctions of the Laplacian on a compact Riemannian manifold which contains an isoparametric submanifold with flat sections to eigenfunctions of the Laplacian of a section. A simple example of such an isoparametric foliation is given by the conjugacy classes of a compact Lie grou…
We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates for stochastically complete submanifolds in Riemannian products, Riemannian submersions and wedges. These estimates are applied for obtaining both horizontal and vertical half-space theorems for submanifolds in $\mathbb{H}^n \times \mathbb{R}^\ell$.
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.
