Locally harmonic manifolds are Riemannian manifolds in which small geodesic spheres are isoparametric hypersurfaces, i.e., hypersurfaces whose nearby parallel hypersurfaces are of constant mean curvature. Flat and rank one symmetric spaces are examples of harmonic manifolds. Damek–Ricci spaces are non-compact harmonic manifolds, most of which are non-symmetric. Taking the limit of an ‘inflating’ sphere through a point p in a Damek–Ricci space as the center of the sphere runs out to infinity along a geodesic half-line $\gamma $ starting from p, we get a horosphere. Similarly to spheres, horospheres are also isoparametric hypersurfaces. In this paper, we define the sphere-like hypersurfaces obtained by ‘overinflating the horospheres’ by pushing the center of the sphere beyond the point at infinity of $\gamma $ along a virtual prolongation of $\gamma $. They give a new family of isoparametric hypersurfaces in Damek–Ricci spaces connecting geodesic spheres to some of the isoparametric hypersurfaces constructed by J. C. Díaz-Ramos and M. Domínguez-Vázquez [17] in Damek–Ricci spaces. We study the geometric properties of these isoparametric hypersurfaces, in particular their homogeneity and the totally geodesic condition for their focal varieties.

In this paper, we prove the non-existence of Codazzi and totally umbilical hypersurfaces, especially totally geodesic hypersurfaces, in the $4$-dimensional model space $\mathrm {Sol}_1^4$.

We prove a local gradient estimate for positive eigenfunctions of the $\mathcal {L}$-operator on conformal solitons given by a general conformal vector field. As an application, we obtain a Liouville type theorem for $\mathcal {L} u=0$, which improves the one of Li and Sun [‘Gradient estimate for the positive solutions of $\mathcal Lu = 0$ and $\mathcal Lu ={\partial u}/{\partial t}$ on conformal solitons’, Acta Math. Sin. (Engl. Ser.) 37(11) (2021), 1768–1782]. We also consider applications where manifolds are special conformal solitons and obtain a better Liouville type theorem in the case of self-shrinkers.

This paper is concerned with the study on an open problem of classifying conformally flat minimal Legendrian submanifolds in the $(2n+1)$-dimensional unit sphere $\mathbb {S}^{2n+1}$ admitting a Sasakian structure $(\varphi,\,\xi,\,\eta,\,g)$ for $n\ge 3$, motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we completely classify such Legendrian submanifolds by assuming that the tensor $K:=-\varphi h$ is semi-parallel, which is introduced as a natural extension of $C$-parallel second fundamental form $h$. Secondly, such submanifolds have also been determined under the condition that the Ricci tensor is semi-parallel, generalizing the Einstein condition. Finally, as direct consequences, new characterizations of the Calabi torus are presented.

We construct inhomogeneous isoparametric families of hypersurfaces with non-austere focal set on each symmetric space of non-compact type and rank ${\geq }3$. If the rank is ${\geq }4$, there are infinitely many such examples. Our construction yields the first examples of isoparametric families on any Riemannian manifold known to have a non-austere focal set. They can be obtained from a new general extension method of submanifolds from Euclidean spaces to symmetric spaces of non-compact type. This method preserves the mean curvature and isoparametricity, among other geometric properties.

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 the present note, we establish a finiteness theorem for $L^p$ harmonic 1-forms on hypersurfaces with finite index, which is an extension of the result of Choi and Seo (J. Geom. Phys. 129 (2018), 125–132).

In this paper, from the property of Killing for structure Jacobi tensor $\mathbb {R}_{\xi }$, we introduce a new notion of cyclic parallelism of structure Jacobi operator $R_{\xi }$ on real hypersurfaces in the complex two-plane Grassmannians. By virtue of geodesic curves, we can give the equivalent relation between cyclic parallelism of $R_{\xi }$ and Killing property of $\mathbb {R}_{\xi }$. Then, we classify all Hopf real hypersurfaces with cyclic parallel structure Jacobi operator in complex two-plane Grassmannians.

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 $m\ge 3$.

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.

First we introduce the notion of parallel Ricci tensor ${\nabla }\mathrm {Ric}=0$ for real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2 and show that the unit normal vector field N is singular. Next we give a new classification of real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2 with parallel Ricci tensor.

We introduce the notion of Lie invariant structure Jacobi operators for real hypersurfaces in the complex quadric $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$. The existence of invariant structure Jacobi operators implies that the unit normal vector field $N$ becomes $\mathfrak{A}$-principal or $\mathfrak{A}$-isotropic. Then, according to each case, we give a complete classification of real hypersurfaces in $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$ with Lie invariant structure Jacobi operators.

In this paper we deal with complete linear Weingarten hypersurfaces immersed into Riemannian space forms. Assuming an Okumura type inequality on the total umbilicity tensor of such hypersurfaces, we prove that either the hypersurface is totally umbilical or it holds an estimate for the norm of the total umbilicity tensor, which is also shown be sharp in the sense that the product of space forms realize them. Our approach is based on a version of the Omori–Yau maximum principle for a suitable Cheng–Yau type operator.

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.

On a real hypersurface $M$ in a complex two-plane Grassmannian ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ we have the Lie derivation $\mathcal{L}$ and a differential operator of order one associated with the generalized Tanaka–Webster connection ${{\widehat{\mathcal{L}}}^{\left( k \right)}}$ . We give a classification of real hypersurfaces $M$ on ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ satisfying $\widehat{\mathcal{L}}_{\xi }^{\left( k \right)}\,S\,=\,{{\mathcal{L}}_{\xi }}S$ , where $\xi$ is the Reeb vector field on $M$ and $s$ the Ricci tensor of $M$ .

We give a classification of Levi-umbilical real hypersurfaces in a complex space form $\widetilde{M}_{n}(c)$ , $n\geqslant 3$ , whose Levi form is proportional to the induced metric by a nonzero constant. In a complex projective plane $\mathbb{C}\mathbb{P}^{2}$ , we give a local construction of such hypersurfaces and moreover, we give new examples of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$ .

There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ . Among them, Suh classified Hopf hypersurfaces in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ with Reeb parallel Ricci tensor in Levi–Civita connection. In this paper, we introduce the notion of generalized Tanaka–Webster $\left( \text{GTW} \right)$ Reeb parallel Ricci tensor for Hopf hypersurfaces in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ . Next, we give a complete classification of Hopf hypersurfaces in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ with $\text{GTW}$ Reeb parallel Ricci tensor.

We study generic and conformally flat hypersurfaces in Euclidean four-space. What kind of conformally flat three manifolds are really immersed generically and conformally in Euclidean space as hypersurfaces? According to the theorem due to Cartan [1], there exists an orthogonal curvature-line coordinate system at each point of such hypersurfaces. This fact is the first step of our study. We classify such hypersurfaces in terms of the first fundamental form. In this paper, we consider hypersurfaces with the first fundamental forms of certain specific types. Then, we give a precise representation of the first and the second fundamental forms of such hypersurfaces, and give exact shapes in Euclidean space of them.

In 1972, Nomizu and Rodriguez [5] found the following characterization of the complete umbilic submanifolds of Euclidean space.

Theorem A. Let Mn, n ≥ 2, be a connected, complete Riemannian manifold isometrically immersed in a Euclidean space Em. Every Morse function of the form Lp has index 0 or n at all of its critical points if and only if Mnis embedded as a Euclidean n-subspace or a Euclidean n-sphere in Em .

We describe all centrioles in irreducible simply connected pointed symmetric spaces of compact type in terms of the root system of the ambient space, and we study some geometric properties of centrioles.