This article introduces the Clairaut conformal Riemannian map. This notion includes the previously studied notions of Clairaut conformal submersion, Clairaut Riemannian submersion, and the Clairaut Riemannian map as particular cases, and is well known in the classical theory of surfaces. Toward this, we find the necessary and sufficient condition for a conformal Riemannian map $\varphi : M \to N$ between Riemannian manifolds to be a Clairaut conformal Riemannian map with girth $s = e^f$. We show that the fibers of $\varphi $ are totally umbilical with mean curvature vector field the negative gradient of the logarithm of the girth function, that is, $-\nabla f$. Using this, we obtain a local splitting of M as a warped product and a usual product, if the horizontal space is integrable (under some appropriate hypothesis). We also provide some examples of the Clairaut conformal Riemannian maps to confirm our main theorem. We observe that the Laplacian of the logarithmic girth, that is, of f, on the total manifold takes the special form. It reduces to the Laplacian on the horizontal distribution, and if it is nonnegative, the universal covering space of M becomes a product manifold, under some hypothesis on f. Analysis of the Laplacian of f also yields the splitting of the universal covering space of M as a warped product under some appropriate conditions. We calculate the sectional curvature and mixed sectional curvature of M when f is a distance function. We also find the relationships between the total manifold and the fibers being symmetrical and, in particular, having constant sectional curvature, and from there, we compare their universal covering spaces, if fibers are also complete, provided f is a distance function. We also find a condition on the curvature tensor of the fibers to be semi-symmetric, provided that the total manifold is semi-symmetric and f is a distance function. In turn, this gives the warped product of symmetric, semi-symmetric spaces into two symmetric, semi-symmetric subspaces (under some hypothesis). Also if the Hessian or the Laplacian of the Riemannian curvature tensor fields is zero, or has a harmonic curvature tensor, then the fibers of $\varphi $ also satisfy the same property, if f is also a distance function. By obtaining Bochner-type formulas for Clairaut conformal Riemannian maps, we establish the relations between the divergences of the Ricci curvature tensor on fibers and horizontal space and the corresponding scalar curvature. We also study the horizontal Killing vector field of constant length and show that they are parallel under appropriate hypotheses. This in turn gives the splitting of the total manifold, if it admits a horizontal parallel Killing vector field and if the horizontal space is integrable. Finally, assuming that $\nabla f$ is a nontrivial gradient Ricci soliton on M, we prove that any vertical vector field is incompressible and hence the volume form of the fiber is invariant under the flow of the vector field.

Li, Ma and Wang have provided in [13] a partial classification of the so-called Moebius deformable hypersurfaces, that is, the umbilic-free Euclidean hypersurfaces $f\colon M^n\to \mathbb{R}^{n+1}$ that admit non-trivial deformations preserving the Moebius metric. For $n\geq 5$, the classification was completed by the authors in [12]. In this article we obtain an infinitesimal version of that classification. Namely, we introduce the notion of an infinitesimal Moebius variation of an umbilic-free immersion $f\colon M^n\to \mathbb{R}^m$ into Euclidean space as a one-parameter family of immersions $f_t\colon M^n\to \mathbb{R}^m$, with $t\in (-\epsilon, \epsilon)$ and $f_0=f$, such that the Moebius metrics determined by ft coincide up to the first order. Then we characterize isometric immersions $f\colon M^n\to \mathbb{R}^m$ of arbitrary codimension that admit a non-trivial infinitesimal Moebius variation among those that admit a non-trivial conformal infinitesimal variation, and use such characterization to classify the umbilic-free Euclidean hypersurfaces of dimension $n\geq 5$ that admit non-trivial infinitesimal Moebius variations.

Let $f\colon M^{2n}\to \mathbb {R}^{2n+p}$, $2\leq p\leq n-1$, be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng (2013, Michigan Mathematical Journal 62, 421–441) conjectured that if the codimension is $p\leq 11$, then, along any connected component of an open dense subset of $M^{2n}$, the submanifold is as follows: it is either foliated by holomorphic submanifolds of dimension at least $2n-2p$ with tangent spaces in the kernel of the second fundamental form whose images are open subsets of affine vector subspaces, or it is embedded holomorphically in a Kaehler submanifold of $\mathbb {R}^{2n+p}$ of larger dimension than $2n$. This bold conjecture was proved by Dajczer and Gromoll just for codimension 3 and then by Yan and Zheng for codimension 4. In this paper, we prove that the second fundamental form of the submanifold behaves pointwise as expected in case that the conjecture is true. This result is a first fundamental step for a possible classification of the nonholomorphic Kaehler submanifolds lying with low codimension in Euclidean space. A counterexample shows that our proof does not work for higher codimension, indicating that proposing $p=11$ in the conjecture as the largest codimension is appropriate.

The local classification of Kaehler submanifolds $M^{2n}$ of the hyperbolic space $\mathbb{H}^{2n+p}$ with low codimension $2\leq p\leq n-1$ under only intrinsic assumptions remains a wide open problem. The situation is quite different for submanifolds in the round sphere $\mathbb{S}^{2n+p}$, $2\leq p\leq n-1$, since Florit et al. [7] have shown that the codimension has to be $p=n-1$ and then that any submanifold is just part of an extrinsic product of two-dimensional umbilical spheres in $\mathbb{S}^{3n-1}\subset\mathbb{R}^{3n}$. The main result of this paper is a version for Kaehler manifolds isometrically immersed into the hyperbolic ambient space of the result in [7] for spherical submanifolds. Besides, we generalize several results obtained by Dajczer and Vlachos [5].

In this paper, we solved an open problem raised by Cecil and Ryan (2015, Geometry of Hypersurfaces, Springer Monographs in Mathematics, Springer, New York, p. 531) by proving the nonexistence of non-Hopf Ricci-semisymmetric real hypersurfaces in $\mathbb {C}P^{2}$ and $\mathbb {C}H^{2}$.

In this paper, by applying for a new approach of the so-called Tsinghua principle, we prove the nonexistence of locally conformally flat real hypersurfaces in both the m-dimensional complex quadric $Q^m$ and the complex hyperbolic quadric $Q^{m\ast }$ for $m\ge 3$ .

We investigate real hypersurfaces in nonflat complex space forms attaining equality in an inequality involving the contact δ-invariant δc(2) introduced by Chen and Mihai in [3].

In this paper we obtain some new characterizations of pseudo-Einstein real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. More precisely, we prove that a real hypersurface in $\mathbb{C}P^{2}$ or $\mathbb{C}H^{2}$ with constant mean curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein. We prove that a real hypersurface in $\mathbb{C}P^{2}$ with constant scalar curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein.

We first provide a necessary and sufficient condition for a ruled real hypersurface in a nonflat complex space form to have constant mean curvature in terms of integral curves of the characteristic vector field on it. This yields a characterization of minimal ruled real hypersurfaces by circles. We next characterize the homogeneous minimal ruled real hypersurface in a complex hyperbolic space by using the notion of strong congruency of curves.

We study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 𝕊3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.

On a real hypersurface $M$ in a non-flat complex space form there exist the Levi–Civita and the $k$-th generalized Tanaka–Webster connections. The aim of this paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operatorwith respect to the Levi–Civita connection coincides with the Lie derivative of it with respect to the $k$-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in direction of any vector field orthogonal to the structure vector field.

We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of ${{\mathbb{S}}^{n}}\,\times \,\mathbb{R}$, extending the classification of umbilical surfaces in ${{\mathbb{S}}^{2}}\times \mathbb{R}$ by Souam and Toubiana as well as the local description of umbilical hypersurfaces in ${{\mathbb{S}}^{n}}\,\times \,\mathbb{R}$ by Van der Veken and Vrancken. We prove that, besides small spheres in a slice, up to isometries of the ambient space they come in a two-parameter family of rotational submanifolds whose substantial codimension is either one or two and whose profile is a curve in a totally geodesic ${{\mathbb{S}}^{1}}\times \mathbb{R}$ or ${{\mathbb{S}}^{2}}\times \mathbb{R}$, respectively, the former case arising in a one-parameter family. All of them are diffeomorphic to a sphere, except for a single element that is diffeomorphic to Euclidean space. We obtain explicit parametrizations of all such submanifolds. We also study more general classes of submanifolds of ${{\mathbb{S}}^{n}}\times \mathbb{R}$ and ${{\mathbb{H}}^{n}}\times \mathbb{R}$. In particular, we give a complete description of all submanifolds in those product spaces for which the tangent component of a unit vector field spanning the factor $\mathbb{R}$ is an eigenvector of all shape operators. We show that surfaces with parallel mean curvature vector in ${{\mathbb{S}}^{n}}\times \mathbb{R}$ and ${{\mathbb{H}}^{n}}\times \mathbb{R}$ having this property are rotational surfaces, and use this fact to improve some recent results by Alencar, do Carmo, and Tribuzy. We also obtain a Dajczer-type reduction of codimension theorem for submanifolds of ${{\mathbb{S}}^{n}}\times \mathbb{R}$ and ${{\mathbb{H}}^{n}}\times \mathbb{R}$.

Let A = ρ2∑i,jAijθi ⊗ θj and B = ρ2∑i,jBij θi ⊗ θj be the Blaschke tensor and the Möbius second fundamental form of the immersion x. Let D = A + λB be the para-Blaschke tensor of x, where λ is a constant. If x: Mn ↦ Sn + 1(1) is an n-dimensional para-Blaschke isoparametric hypersurface in a unit sphere Sn + 1(1) and x has three distinct Blaschke eigenvalues one of which is simple or has three distinct Möbius principal curvatures one of which is simple, we obtain the full classification theorems of the hypersurface.

We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time.

Let Mn be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1(1), n ≤ 8 and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and .

We extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying that an isometric immersion is k-umbilical when APk−1(A) is a multiple of the identity, where Pk(A) is the kth Newton polynomial in the second fundamental form A with P0(A) being the identity. Thus, for k=1, one-umbilical coincides with umbilical. We determine the principal curvatures of the two-umbilical isometric immersions in terms of the mean curvatures. We give a description of the two-umbilical isometric immersions in space forms which includes the product of spheres embedded in the Euclidean sphere S2k+1 of radius 1. We also introduce an operator φk which measures how an isometric immersion fails to be k-umbilical, giving in particular that φ1 ≡ 0 if and only if the immersion is totally umbilical. We characterize the two-umbilical hypersurfaces of a space form as images of isometric immersions of Einstein manifolds.

We give a formula for the Laplacian of the second fundamental form of an n-dimensional compact minimal submanifold M in a complex projective space CPm. As an application of this formula, we prove that M is a geodesic minimal hypersphere in CPm if the sectional curvature satisfies K≥1/n, if the normal connection is flat, and if M satisfies an additional condition which is automatically satisfied when M is a CR submanifold. We also prove that M is the complex projective space CPn/2 if K≥3/n, and if the normal connection of M is semi-flat.

In this paper, we characterize hypersurfaces of type A2 in a complex projective space in terms of their geodesics.

We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is of Codazzi type.

We provide a complete local classification of pseudo-parallel sub-manifolds with flat normal bundle of space forms, extending the classification by Dillen-Nölker for the semi-parallel case.