In this paper, we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call almost quotient Yamabe solitons as they extend quite naturally those already called quotient Yamabe solitons. We present sufficient conditions for a compact almost quotient Yamabe soliton to be either trivial or isometric with an Euclidean sphere. We also characterize noncompact almost gradient quotient Yamabe solitons satisfying certain conditions on both its Ricci tensor and potential function.

We establish a link between the behavior of length functions on Teichmüller space and the geometry of certain anti-de Sitter $3$-manifolds. As an application, we give new purely anti-de Sitter proofs of results of Teichmüller theory such as (strict) convexity of length functions along shear paths and geometric bounds on their second variation along earthquakes. Along the way, we provide shear-bend coordinates for GHMC anti-de Sitter $3$-manifolds.

In a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric, and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the existence of null hypersurface with equal nontrivial principal curvatures, and finally, we give a necessary condition on the ambient scalar curvature, for the existence of null (non-minimal) hypersurfaces that are of constant mean curvature.

We investigate geometric aspects of complete spacelike mean curvature flow solitons of codimension one in a generalized Robertson–Walker (GRW) spacetime $-I\times _{f}M^n$, with base $I\subset \mathbb R$, Riemannian fiber $M^n$ and warping function $f\in C^\infty (I)$. For this, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space and to obtain nonexistence results concerning these solitons. In particular, we deal with entire graphs constructed over the Riemannian fiber $M^n$, which are spacelike mean curvature flow solitons, and we also explore the geometry of a conformal vector field to establish topological and further rigidity results for compact (without boundary) mean curvature flow solitons in a GRW spacetime. Moreover, we study the stability of spacelike mean curvature flow solitons with respect to an appropriate stability operator. Standard examples of spacelike mean curvature flow solitons in GRW spacetimes are exhibited, and applications related to these examples are given.

Pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as essentially conformally symmetric (ECS) manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the rank of a certain distinguished null parallel distribution $\mathcal{D}$. All known examples of compact ECS manifolds are of rank one and have dimensions greater than 4. We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced when necessary by a twofold isometric covering, must be a bundle over the circle with leaves of $\mathcal{D}^\perp$ serving as the fibres. The same conclusion holds in the locally homogeneous case if one assumes that $\,\mathcal{D}^\perp$ has at least one compact leaf. We also show that in the pseudo-Riemannian universal covering space of any compact rank-one ECS manifold, the leaves of $\mathcal{D}^\perp$ are the factor manifolds of a global product decomposition.

Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz–Killing curvature integrals of a submanifold with corners, to be computed by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

We determine all three-dimensional homogeneous and $1$-curvature homogeneous Lorentzian metrics which are critical for a quadratic curvature functional. As a result, we show that any quadratic curvature functional admits different non-Einstein homogeneous critical metrics and that there exist homogeneous metrics which are critical for all quadratic curvature functionals without being Einstein.

By means of a counter-example, we show that the Reilly theorem for the upper bound of the first non-trivial eigenvalue of the Laplace operator of a compact submanifold of Euclidean space (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) does not work for a (codimension ⩾2) compact spacelike submanifold of Lorentz–Minkowski spacetime. In the search of an alternative result, it should be noted that the original technique in (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) is not applicable for a compact spacelike submanifold of Lorentz–Minkowski spacetime. In this paper, a new technique, based on an integral formula on a compact spacelike section of the light cone in Lorentz–Minkowski spacetime is developed. The technique is genuine in our setting, that is, it cannot be extended to another semi-Euclidean spaces of higher index. As a consequence, a family of upper bounds for the first eigenvalue of the Laplace operator of a compact spacelike submanifold of Lorentz–Minkowski spacetime is obtained. The equality for one of these inequalities is geometrically characterized. Indeed, the eigenvalue achieves one of these upper bounds if and only if the compact spacelike submanifold lies minimally in a hypersphere of certain spacelike hyperplane. On the way, the Reilly original result is reproved if a compact submanifold of a Euclidean space is naturally seen as a compact spacelike submanifold of Lorentz–Minkowski spacetime through a spacelike hyperplane.

We prove the existence of compact spacelike hypersurfaces with prescribed k-curvature in de Sitter space, where the prescription function depends on both space and the tilt function.

We prove that among all compact homogeneous spaces for an effective transitive action of a Lie group whose Levi subgroup has no compact simple factors, the seven-dimensional flat torus is the only one that admits an invariant torsion-free $\text{G}_{2(2)}$-structure.

Following an original idea of Palmas, Palomo and Romero, recently developed in [12], we study codimension two spacelike submanifolds contained in the light cone of the Lorentz-Minkowski spacetime through an approach which allows us to compute their extrinsic and intrinsic geometries in terms of a single function u. As the first application of our approach, we classify the totally umbilical ones. For codimension two compact spacelike submanifolds into the light cone, we show that they are conformally diffeomorphic to the round sphere and that they are given by an explicit embedding written in terms of u. In the last part of the paper, we consider the case where the submanifold is (marginally, weakly) trapped. In particular, we derive some non-existence results for weakly trapped submanifolds into the light cone.

We prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension n = 2. This then implies that the mean curvature flow exists for all time and converges to a translating solution.

We study space-like hypersurfaces with functionally bounded mean curvature in Lorentzian warped products , where F is a (non-compact) complete Riemannian manifold whose universal covering is parabolic. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. As an application, several Calabi–Bernstein-type results are obtained which widely extend the previous ones in this setting.

In this article, we study complete surfaces $\sum $, isometrically immersed in the product spaces ${{\mathbb{H}}^{2}}\,\times \,\mathbb{R}$ or ${{\mathbb{S}}^{2\,}}\times \,\mathbb{R}$ having positive extrinsic curvature ${{K}_{e}}$. Let ${{K}_{i}}$ denote the intrinsic curvature of $\sum $. Assume that the equation $a{{K}_{i\,}}\,+\,b{{K}_{e\,}}\,=\,c$ holds for some real constants $a\,\ne \,0$, $b\,>\,0$, and $c$. The main result of this article states that when such a surface is a topological sphere, it is rotational.

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

In this paper, we establish new characterization results concerning totally umbilical hypersurfaces of the hyperbolic space $\mathbb{H}^{n+1}$, under suitable constraints on the behavior of the Lorentzian Gauss map of complete hypersurfaces having some constant higher order mean curvature. Furthermore, working with different warped product models for $\mathbb{H}^{n+1}$ and supposing that certain natural inequalities involving two consecutive higher order mean curvature functions are satisfied, we study the rigidity and the nonexistence of complete hypersurfaces immersed in $\mathbb{H}^{n+1}$.

In this paper we study the Lorentzian surfaces with finite type Gauss map in the four-dimensional Minkowski space. First, we obtain the complete classification of minimal surfaces with pointwise 1-type Gauss map. Then, we get a classification of Lorentzian surfaces with nonzero constant mean curvature and of finite type Gauss map. We also give some explicit examples.

In this paper we prove two inequalities relating the warping function to various curvature terms, for warped products isometrically immersed in Riemannian manifolds. This extends work by B. Y. Chen [‘On isometric minimal immersions from warped products into real space forms’, Proc. Edinb. Math. Soc. (2) 45(3) (2002), 579–587 and ‘Warped products in real space forms’, Rocky Mountain J. Math.34(2) (2004), 551–563] for the case of immersions into space forms. Finally, we give an application where the target manifold is the Clifford torus.

In this paper we characterize $K$-contact semi-Riemannian manifolds and Sasakian semi-Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat $K$-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature $\kappa \,=\,\varepsilon$, where $\varepsilon \,=\,\pm 1$ denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a $K$-contact Lorentzian manifold.

We study the geometry of non-reductive four-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.