Let $M$ be a finite volume analytic pseudo-Riemannian manifold that admits an isometric $G$-action with a dense orbit, where $G$ is a connected non-compact simple Lie group. For low-dimensional $M$, i.e. $\dim(M) < 2\dim(G)$, when the normal bundle to the $G$-orbits is non-integrable and for suitable conditions, we prove that $M$ has a $G$-invariant metric which is locally isometric to a Lie group with a bi-invariant metric (local rigidity theorem). The latter does not require $M$ to be complete as in previous works. We also prove a general result showing that $M$ is, up to a finite covering, of the form $H/Γ$ ($Γ$ a lattice in the group $H$) when we assume that $M$ is complete (global rigidity theorem). For both the local and the global rigidity theorems we provide cases that imply the rigidity of $G$-actions for $G$ given by $\mathrm{SO}_0(p,q)$, $G_{2(2)}$ or a non-compact simple Lie group of type $F_4$ over $\mathbb{R}$. We also survey the techniques and results related to this work.
In this paper, we show the rigidity of isometric immersions for a Riemannian manifold of dimension $n-1$ into the light cone of $n+1$ dimensional Minkowski, de Sitter and anti-de Sitter spacetimes for $n\geq 3$.
We show that among the Euclidean submanifolds with codimension two the ones of rank two that are parabolic but nonruled are isometrically rigid. This generalizes the result in [10] that these submanifolds are genuinely rigid. In addition, we give a parametric classifications of all parabolic submanifolds.
We characterize the universal covering of connected analytic pseudo-Riemannian manifolds which admit a non-trivial and isometric action of the simple Lie group $SL(3,\mathbb{R})$ with a dense orbit preserving a finite volume. If such manifold is also weakly irreducible we prove that $M$ is isometric to, or a quotient space of, a simple Lie group containing $SL(3,\mathbb{R})$.
Let $(M,g)$ be a compact manifold with boundary and $Ric_g\geq (n-1)g$, Hang and Wang proved that $(M,g)$ is isometric to the standard hemisphere if $\partial M$ is convex and isometric to $\mathbb{S}^{n-1}(1)$. We prove some rigidity theorems when $\partial M $ is isometric to a product manifold where one factor is the standard sphere.
In this paper, we study biharmonic isometric immersions of a surface into and biharmonic Riemannian submersion from 3-dimensional Berger spheres. We obtain a classification of proper biharmonic isometric immersions of a surface with constant mean curvature into Berger 3-spheres. We also give a complete classification of proper biharmonic Hopf tori in Berger 3-sphere. For Riemannian submersions, we prove that a Riemannian submersion from Berger 3-spheres into a surface is biharmonic if and only if it is harmonic.
We analyze sub-Riemannian and lightlike metrics from the point of view of their rigidity as geometric structures. Following Cartan's and Gromov's formal definitions, they are never rigid, yet, in generic cases, they naturally give rise to rigid geometric structures!?
We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by M. do Carmo and E. Lima, as well as by M. do Carmo and F. Warner. It also settles affirmatively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second funda…
We prove the following rigidity theorem: For an n-dimensional compact Riemannian manifold with boundary whose Ricci curvature is bounded by n-1 from below, if its boundary is isometric to the standard sphere of dimension n-1 and totally geodesic, then the manifold is isometric to the standard hemisphere.
Several characterizations of umbilic points of submanifolds in arbitrary Riemannian and Lorentzian manifolds are given. As a consequence, we obtain new characterizations of spheres in the Euclidean space and of hyperbolic spaces in the Lorentz-Minkowski space. We also prove the Lorentzian version of a classical result by Cartan.
We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove geometrical inequalities for a sub-Riemannian submanifold. As applications, inequalities are proved for submanifolds with mutually orthogonal distributions that include scalar and mutual curvature. For compact submanifolds, inequalities are obtained that are supported by known integral formulas for almost-product manifolds.
This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannain situation; for instance the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with left-invariant pseudo-Riemannian metric we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo-$H$-type groups.
For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant.
We prove that the Euclidean plane is the only Riemannian plane with total curvature and free of conjugate points that satisfies Playfair's version of the parallel postulate.
Let $M$ be a Riemannian manifold with dimension greater or equal to $3$ which admits a complete, finite-volume Riemannian metric $g_0$ locally isometric to a rank-1 symmetric space of non-compact type. The volume entropy rigidity theorem asserts that $g_0$ minimizes a normalized volume growth entropy among all complete, finite-volume, Riemannian metric on $M$. We will repair a gap in the proof when $g_0$ is locally isometric to the Cayley hyperbolic space.
We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields on the manifold, satisfying a certain non-linear equation involving the Riemannian curvature tensor of $M$. Setting $k=1$, we proceed to recover the fundamental theorem of hypersurfaces. In the case of manifolds of positive sectional curvature and $n\geq 3$, we reduce the solvability of the Gauss and Codazzi equations to the cancelation of a set of obstructio…
This note is concerned with some essential properties (optimal isoperimetry, first variation, and monotonicity formula) of the so-called $[0,1)\niγ$-torsional rigidity $\mathcal{T}_{γ,\mathsf{g}}$ on a complete Riemannian two-manifold $(\mathbb M^2,\mathsf{g})$. Even in the special case of $\mathbb R^2$, major results are new.
Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there is no almost-isometry between the universal covers. We show that Riemannian manifolds which are almost-isometric have the same volume growth entropy. We establish various rigidity results as applications.
We prove the developability and $C^{1,1/2}$ regularity of $W^{2,2}$ isometric immersions of $n$-dimensional domains into $R^{n+1}$. As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the $W^{2,2}$ strong norm, provided the domain is $C^1$ and convex. Both results fail to be true if the Sobolev regularity is weaker than $W^{2,2}$.
In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.
For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Yamabe invariant, we also provide a similar rigidity result. As an application, we obtain some rigidity results on 4-dimensional manifolds.
A Riemannian manifold endowed with $k\ge2$ complementary pairwise orthogonal distributions is called a Riemannian almost $k$-product manifold. In the article, for the first time, we study the following problem: find a relationship between intrinsic and extrinsic invariants of a Riemannian almost $k$-product manifold isometrically immersed in another Riemannian manifold. For such immersions, we establish an optimal inequality that includes the mixed scalar curvature and the square of the mean curvature. Although Riemannian curvature tensor belongs to intrinsic geometry, a special part called th…
In this paper, we show that every harmonic map from a compact Kähler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact Kähler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.
In this paper we prove a rigidity result for the equality case of the Penrose inequality on $3$-dimensional asymptotically flat manifolds with nonnegative scalar curvature and corners. Our result also has deep connections with the equality cases of Theorem 1 in \cite{Miao2} and Theorem 1.1 in \cite{LM}.
We study Riemannian foliations with complex leaves on Kaehler manifolds. The tensor T, the obstruction to the foliation be totally geodesic, is interpreted as a holomorphic section of a certain vector bundle. This enables us to give classification results when the manifold is compact.
In this article, we prove a rigidity theorem for isometric embeddings into the Schwarzschild manifold, by using the variational formula of quasi-local mass.
In this article, we prove new rigidity results for compact Riemannian spin manifolds with boundary whose scalar curvature is bounded from below by a non-positive constant. In particular, we obtain generalizations of a result of Hang-Wang \cite{hangwang1} based on a conjecture of Schroeder and Strake \cite{schroeder}.
Due to Janet-Cartan's theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M,g), it is in general hard to determine the least dimensional Euclidean space into which (M,g) can be locally isometrically embedded, even in the case where (M,g) is homogeneous. In this paper, when the space (M,g) is locally isometric to a three-dimensional Lie group equipped with a left-invariant Riemannian metric, we classify all such spaces that can be locally isometrically embedded into…
We prove here that given a proper isometric action $K\times M\to M$ on a complete Riemannian manifold $M$ then every continuous isometric flow on the orbit space $M/K$ is smooth, i.e., it is the projection of an $K$-equivariant smooth flow on the manifold $M$. As a direct corollary we infer the smoothness of isometric actions on orbit spaces. Another relevant application of our result concerns Molino's conjecture, which states that the partition of a Riemannian manifold into the closures of the leaves of a singular Riemannian foliation is still a singular Riemannian foliation. We prove Molino'…
We prove rigidity of oriented isometric immersions of complete surfaces in the homo- geneous 3-manifolds E(k; τ) (different from the space forms) having the same positive extrinsic curvature.
Let $M$ be a compact connected pseudo-Riemannian manifold on which a solvable connected Lie group $G$ of isometries acts transitively. We show that $G$ acts almost freely on $M$ and that the metric on $M$ is induced by a bi-invariant pseudo-Riemannian metric on $G$. Furthermore, we show that the identity component of the isometry group of $M$ coincides with $G$.
Let $(M, \bar g)$ be an $n$-dimensional complete Riemannian manifold. In this paper, we considers the following conformal scalar curvature rigidity problem: Given a compact smooth domain $Ω$ with $\partial Ω$, can one find a conformal metric $g$ whose scalar curvature $R[g]\ge R[\bar g]$ on $Ω$ and the mean curvature $H[g] \ge H[ \bar g]$ on $\partial Ω$ with $\bar g = g$ on $\partial Ω$? We prove that $\bar g = g$ on some smooth domains in a general Riemannian manifold, which is an extension of the previous results given by Qing and Yuan, and Hang and Wang.
Consider a compact Riemannian manifold with boundary endowed with a magnetic field. A path taken by a particle of unit charge, mass, and energy is called a magnetic geodesic. It is shown that if everything is real-analytic, the topology, metric, and magnetic field are uniquely determined by the scattering relation of the magnetic geodesic flow, measured at the boundary. Conjugate points are allowed with minor restrictions. In exchange for the real-analytic assumption, prior results in boundary rigidity are greatly generalized.
We derive general structure and rigidity theorems for submetries $f: M \to X$, where $M$ is a Riemannian manifold with sectional curvature $\sec M \ge 1$. When applied to a non-trivial Riemannian submersion, it follows that $diam X \leq π/2 $. In case of equality, there is a Riemannian submersion $\mathbb{S} \to M$ from a unit sphere, and as a consequence, $f$ is known up to metric congruence. A similar rigidity theorem also holds in the general context of Riemannian foliations.
We prove that a Kahler manifold with positive bisectional curvature and maximal diameter is isometric to the complex projective space with the Fubini-Study metric.
In this paper we provide two new characterizations of real hyperbolic $n$-space using the Poincaré exponent of a discrete group and the volume growth entropy. The first characterization is in the space of Hilbert metrics and generalizes a result of Crampon. The second is in the space of Riemannian metrics with Ricci curvature bounded below and generalizes a result of Ledrappier and Wang.
We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div-curl lemma can be employed to establish the global weak rigidity of the Gauss-Codazzi-Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.
In this paper we present a rigidity theorem for locally isometric hypersurfaces with a curvature restriction in de Sitter space. This is an analogue to the case for Riemannian space forms given by Guan and Shen in [5].
We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several classes of isometric actions, including polar and variationally complete ones. All results are proven in the more general case of singular Riemannian foliations.
We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact $2$-Riem…
Let $Γ$ be a discrete group with property $(T)$ of Kazhdan. We prove that any Riemannian isometric action of $Γ$ on a compact manifold $X$ is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in \cite{FM2}. One definition of property $(T)$ is that a group $Γ$ has property $(T)$ if every isometric $Γ$ action on a Hilbert space has a fixed point. We prove a variety of strengthenings of this fixed point properties for groups with proper…
Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=π. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and positive spherical rank, then M is isometric to a compact, rank one symmetric space (CROSS) i.e., isometric to a sphere, complex projective space, quaternionic projective space or to the Cayley plane. The notion of spherical rank is analogous to the notions of Euclidean rank and hy…
We show that if the structure algebra of a Riemannian foliation F on a closed manifold M is nilpotent, then the integral of the Álvarez class of (M,F) along every closed path is the exponential of an algebraic number. By this result and the continuity of the Álvarez class under deformations shown in arXiv:1009.1098v2, we prove that the Álvarez class and the geometrically tautness of Riemannian foliations on a closed manifold M are invariant under deformation, if the fundamental group of M has polynomial growth.
The purpose of this article is to prove that, under suitable constrains on the electrovacuum spacetime $M$, if $Σ\subset M$ is an embedded strictly stable minimal two-sphere which locally maximizes the charged Hawking mass, then there exist a neighborhood of it in $M$ isometric to the Reissner-Nordström-de Sitter space. At the same time, motived by \cite{Brendle}, we will deduce an estimate for area of a two-sphere which is locally area minimizing in an electrovacuum spacetime. Moreover, if the equality holds, then there exist a neighborhood of it in $M$ isometric to the charged Nariai space.
Here, an extension of the Obata-Tanno's theorem to Finsler geometry is established and the following rigidity result is obtained; Every complete connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to an $n$-sphere equipped with a certain Finsler metric, and vise versa.
