Miguel Dominguez-Vazquez, Olga Perez-Barral
We complete the classification of ruled real hypersurfaces with shape operator of constant norm in nonflat complex space forms by showing the existence of a unique inhomogeneous example in the complex hyperbolic space.
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We study ruled real hypersurfaces whose shape operators have constant squared norm in nonflat complex space forms. In particular, we prove the nonexistence of such hypersurfaces in the projective case. We also show that biharmonic ruled real hypersurfaces in nonflat complex space forms are minimal, which provides their classification due to a known result of Lohnherr and Reckziegel.
We show that ruled real hypersurfaces with constant mean curvature in the complex projective and hyperbolic spaces must be minimal. This provides their classification, by virtue of a result of Lohnherr and Reckziegel.
Geometry of pseudo-non-degenerate two-ruled hypersurfaces
本文研究了双 rulings 超曲面的几何特性，提出伪非退化定义并分析其性质，通过高度函数构造伪非退化超曲面并探讨其奇点与原始曲线的关系。
The main two families of real hypersurfaces in complex space forms are Hopf and ruled. However, very little is known about real hypersurfaces in the indefinite complex projective space $\cpn$. In a previous work, Kimura and the second author introduced Hopf real hypersurfaces in $\cpn$. In this paper, ruled real hypersurfaces in the indefinite complex projective space are introduced, as those whose maximal holomorphic distribution is integrable, and such that the leaves are totally geodesic holomorphic hyperplanes. A detailed description of the shape operator is computed, obtaining two main di…
2-Ruled Hypersurfaces in a Walker 4-Manifold
本文在Walker 4-流形中定义了三种类型的2-直纹超曲面，计算了其高斯曲率和平均曲率，刻画了极小性，并处理了第一Laplace-Beltrami算子。
In this paper, we define three types of 2-Ruled hypersurfaces in the Minkowski 4-space $\mathbb{E}^4_1$. We obtain Gaussian and mean curvatures of the 2-ruled hypersurfaces of type-1 and type-2, and some characterizations about its minimality. We also deal with the first Laplace-Beltrami operators of these types of 2-Ruled hypersurfaces in $\mathbb{E}^4_1$. Moreover, the importance of this paper is that the definition of these surfaces by using the octonions in $\mathbb{E}^4_1$. Thus, this is a new idea and make the paper original. We give an example of 2-ruled hypersurface constructed by octo…
We consider real hypersurfaces $M$ in complex projective space equipped with both the Levi-Civita and generalized Tanaka-Webster connections. For any nonnull constant $k$ and any symmetric tensor field of type (1,1) $L$ on $M$ we can define two tensor fields of type (1,2) on $M$, $L_F^{(k)}$ and $L_T^{(k)}$, related to both connections. We study the behaviour of the structure operator $ϕ$ with respect to such tensor fields in the particular case of $L=A$, the shape operator of $M$, and obtain some new characterizations of ruled real hypersurfaces in complex projective space.
Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of $2n$-dimensional nondegenerate hypersurfaces ruled by $n$-planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with respect to which the ruling is Lagrangian.
We construct examples of inhomogeneous isoparametric real hypersurfaces in complex hyperbolic spaces.
Cohomogeneity-One Ruled Hypersurfaces in $\mathbb{CP}^2$ and $\mathbb{C}H^2$
本文通过在全测地一维子流形上构造正交规则，给出了非平坦复空间形式 $\mathbb{CP}^n$ 和 $\mathbb{C}H^n$ 中一类特殊规则超曲面的构造方法，并针对 $n=2$ 情况，证明了当基曲线具有常测地曲率时，该构造恰好得到满足特定横截条件的实解析等秩一超曲面。
We consider hypersurfaces in the real Euclidean space $\mathbb{R}^{n+1}$ ($n\geq2$) which are relatively normalized. We give necessary and sufficient conditions a) for a surface of negative Gaussian curvature in $\mathbb{R}^3$ to be ruled, b) for a hypersurface of positive Gaussian curvature in $\mathbb{R}^{n+1}$ to be a hyperquadric and c) for a relative normalization to be constantly proportional to the equiaffine normalization.
We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces.
In this paper, we study the special curves and ruled surfaces on helix hypersurface whose tangent planes make a constant angle with a fixed direction in Euclidean n-space Besides, we observe some special ruled surfaces in and give requirement of being developable of the ruled surface. Also, we investigate the helix surface generated by a plane curve in Euclidean 3-space .
First, we classify proper biharmonic Hopf real hypersurfaces in $\mathbb{C}P^2$. Next, we classify proper biharmonic real hypersurfaces with two distinct principal curvatures in $\mathbb{C}P^n$, where $n\geq 2$. Finally, we prove that biharmonic ruled real hypersurfaces in $\mathbb{C}P^n$ are minimal, where $n\geq 2$.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In this paper, we study intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures, and sectional curvatures.
We classify hypersurfaces of the Minkowski space $Ł^{n+1}$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or three dimensional strip. Moreover, if the hypersurface is embedded then it is a partial tube over a curve.
On skew loops, skew branes and quadratic hypersurfaces
斜膜是仿射空间中的浸入余维2子流形，没有平行的切空间对。利用莫尔斯理论，我们证明了斜膜不可能位于二次超曲面上.我们还证明了3-空间中嵌入直纹可展圆盘上不存在斜环。本文推广了M. Ghomi和B.所罗门
We construct uncountably many isoparametric families of hypersurfaces in Damek-Ricci spaces. We characterize those of them that have constant principal curvatures by means of the new concept of generalized Kahler angle. It follows that, in general, these examples are inhomogeneous and have nonconstant principal curvatures. We also find new cohomogeneity one actions on quaternionic hyperbolic spaces, and an isoparametric family of inhomogeneous hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane.
Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c\neq 0$. Suppose that the structure vector field $ξ$ of $M$ is an eigen vector field of the Ricci tensor $S$, $Sξ=βξ$, $β$ being a function. We study on $M$, a gradient pseudo-Ricci soliton as an extended concept of Ricci soliton, closely related to pseudo-Einstein real hypersurfaces. We show that a $3$-dimensional ruled real hypersurface of $M^2(c), c<0$ admits a non-trivial gradient pseudo-Ricci soliton.
We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to open parts of tubes around the ruled minimal submanifolds with totally real normal bundle introduced by Berndt and Bruck. In particular, they are open parts of homogenous ones.
Concircular hypersurfaces and concircular helices in space forms
本文研究了空间形式中的圆柱曲面和圆柱螺旋线，通过微分方程描述其性质，并证明圆柱螺旋线是圆柱曲面的测地线。
It is well known that the only surfaces that are simultaneously minimal in $\mathbb{R}^3$ and maximal in $\mathbb{L}^3$ are open pieces of helicoids (in the region in which they are spacelike) and of spacelike planes (O. Kobayashi 1983). The proof of this result consists in showing that the level curves of those surfaces are lines, and so the surfaces are ruled. And it finishes comparing the classification of minimal ruled surfaces to that of maximal ruled surfaces. In this manuscript we consider the general case of spacelike hypersurfaces in the $(n+1)$-dimensional Euclidean space which are s…
A local description of the non-flat infinitesimally bendable Euclidean hypersurfaces was recently given by Dajczer and Vlachos \cite{DaVl}. From their classification, it follows that there is an abundance of infinitesimally bendable hypersurfaces that are not isometrically bendable. In this paper we consider the case of complete hypersurfaces $f\colon M^n\to\mathbb{R}^{n+1}$, $n\geq 4$. If there is no open subset where $f$ is either totally geodesic or a cylinder over an unbounded hypersurface of $\mathbb{R}^4$, we prove that $f$ is infinitesimally bendable only along ruled strips. In particul…
Null hypersurfaces as wave fronts in Lorentz-Minkowski space
本文证明洛伦兹-闵可夫斯基空间中L-完备零超曲面作为波前可由欧几里得空间超曲面典范诱导，并刻画了奇异集为紧集的此类波前。
Classification of ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia
本文研究Lorentz-Minkowski空间中关于原点惯性矩平稳的超曲面，在$\mathbb{L}^3$中分类平稳直纹面，证明平面是唯一平稳柱面，非柱面分类取决于直母线因果特征并给出显式参数化。
We classify 4-dimensional austere submanifolds in Euclidean space ruled by 2-planes. The algebraic possibilities for second fundamental forms of an austere 4-fold M were classified by Bryant, falling into three types which we label A, B, and C. We show that if M is 2-ruled of Type A, then the ruling map from M into the Grassmannian of 2-planes in R^n is holomorphic, and we give a construction for M starting with a holomorphic curve in an appropriate twistor space. If M is 2-ruled of Type B, then M is either a generalized helicoid in R^6 or the product of two classical helicoids in R^3. If M is…
Inhomogeneous Cosmological Models with Homogeneous Inner Hypersurface Geometry
时空允许切片到均匀的空间超曲面推广了通常的Bianchi模型。我们已经知道，在这些模型中，比安基类型可能会随着时间而变化。在这里，我们展示了哪些变化真正出现。为此，我们刻画了拓扑空间的点是三维定向齐次黎曼流形，局部等距流形被认为是相同的。
Inhomogeneity of Isoparametric Hypersurfaces of OT-FKM-type in the Pseudo-Sphere
该研究针对伪黎曼空间形式中的等参超曲面，显式构造任意符号的Clifford系统，并证明满足特定符号条件的伪球面OT-FKM型等参超曲面及其焦集M₊的连通分支均具有非齐性。
Let $M$ be a real hypersurface of a complex space form with constant curvature $c$. In this paper, we study the hypersurface $M$ admitting Miao-Tam critical metric, i.e. the induced metric $g$ on $M$ satisfies the equation:$-(Δ_gλ)g+\nabla^2_gλ-λRic=g$, where $λ$ is a smooth function on $M$. At first, for the case where $M$ is Hopf, $c=0$ and $c\neq0$ are considered respectively. For the non-Hopf case, we prove that the ruled real hypersurfaces of non-flat complex space forms do not admit Miao-Tam critical metrics. Finally, it is proved that a compact hypersurface of a complex Euclidean space …
Characterizations of umbilical hypersurfaces by partially overdetermined problems in space forms
研究空间形式中 Serrin 型部分超定问题对脐超曲面的刚性刻画，将齐次 Robin 边界条件推广为非齐次，从而把接触角从正交推广到任意 $\theta\in(0,\pi)$。
We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \subset N,$ of class $C^{2, α}$ for any $α\in (0,1),$ such that $M$ is the image of a two-sided immersion whose mean curvature is given by $gν$ for an appropriate choice of continuous unit normal $ν$ to the immersion; and moreover, the singular set $Σ= \overline{M} \setminus M$ is empty if $2 \leq n \leq 6,$ finite if $n=7$ and satisfies ${\mathcal H}^{n-7 + γ}(Σ) = 0$ for every $γ>0$ if $n \geq 8$.…
Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank
本文提出一种将子流形从欧几里得空间扩展到非紧型对称空间的新方法，构造出高秩对称空间中具有非严峻焦集的首批非齐次等参超曲面族。
Rigidity of stable spacelike capillary hypersurfaces in de Sitter and Minkowski spaces
该研究证明了德西特与闵可夫斯基空间中，当支撑为非负内蕴曲率的类时全脐超曲面时，保体积稳定的紧致类时毛细管超曲面必为全脐，还推导了统一闵可夫斯基型公式并构造了测试族。
Hypersurfaces with Constant Ricci Eigenvalues in Real Space Forms
本文证明实空间形式中连通超曲面具有常Ricci特征值当且仅当曲率齐性，从而完成分类，推广了Einstein超曲面的分类，并得到曲率非齐性流形不能等距浸入余维一实空间形式的推论。
The classification work [5], [9] left unsettled only those anomalous isoparametric hypersurfaces with four principal curvatures and multiplicity pair $\{4,5\},\{6,9\}$ or $\{7,8\}$ in the sphere. By systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypersurfaces, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $\{4,5\}$ in $S^{19}$ is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities $\{6,9\}$ in $S^{31}$ is either the inhomogen…
In this article, we construct a new para-Kähler structure $({\mathcal G},{\mathcal J},Ω)$ in the space of oriented geodesics ${\mathbb L}(M)$ in a non-flat, real space form $M$. We first show that the para-Kähler metric ${\mathcal G}$ is scalar flat and when $M$ is a 3-dimensional real space form, ${\mathcal G}$ is locally conformally flat. Furthermore, we prove that the space of oriented geodesics in hyperbolic $n$-space, equipped with the constructed metric ${\mathcal G}$, is minimally isometric embedded in the tangent bundle of the hyperbolic $n$-space. We then study the submanifold theory,…
We classify all complete projective special real manifolds with reducible cubic potential, obtaining four series. For two of the series the manifolds are homogeneous, for the two others the respective automorphism group acts with co-homogeneity one. Complete projective special real manifolds give rise to complete quaternionic Kähler manifolds via the supergravity q-map, which is the composition of the supergravity c-map and r-map. We develop curvature formulas for manifolds in the image of the q-map. Applying the q-map to one of the above series of projective special real manifolds, we obtain …
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