Flat Normal Bundle Research Articles

Submanifolds with constant scalar curvature in space forms

Let [Formula: see text] be an [Formula: see text]-dimensional oriented compact submanifold with constant normalized scalar curvature [Formula: see text] in the space form [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the mean curvature and the Ricci curvature of [Formula: see text] respectively. By applying Cheng-Yau’s self-adjoint operator, we first prove that if [Formula: see text] is a hypersurface in a unit sphere, and [Formula: see text], then [Formula: see text] is totally umbilical. Furthermore, we investigate the submanifolds in [Formula: see text] with flat normal bundle satisfying [Formula: see text], and obtain a complete classification theorem.

Vanishing Theorems for p-Harmonic Forms on Submanifolds in Spheres

In this paper, we present some vanishing theorems for p-harmonic forms on -super stable complete submanifold M immersed in sphere Sn + m. When 2 ≤ 1 ≤ n-2, M has a flat normal bundle. Assuming that M is a minimal submanifold and δ > 1(n–1)p2 / 4n[p–1+(p–1)2kp], we prove a vanishing theorem for p-harmonic ℓ-forms.

Submanifolds with constant Moebius curvature and flat normal bundle

Submanifolds with constant Moebius curvature and flat normal bundle

$L$-parabolic linear Weingarten spacelike submanifolds immersed in an Einstein manifold

In this paper, we deal with complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector and flat normal bundle in an Einstein manifold \smash{\mathcal E^{n+p}_p} of index p . Under some curvature constraints on the ambient space, we establish an L -parabolicity criterion related to a suitable modified Cheng–Yau's operator L and we apply it to obtain sufficient conditions which guarantee that such a spacelike submanifold must be an isoparametric hypersurface of \smash{\mathcal E^{n+1}_1} with two distinct principal curvatures one of which is simple.

On the complement of a hypersurface with flat normal bundle which corresponds to a semipositive line bundle

We investigate the complex analytic structure of the complement of a non-singular hypersurface with unitary flat normal bundle when the corresponding line bundle admits a Hermitian metric with semipositive curvature.

Isometric immersions with flat normal bundle between space forms

We investigate the behavior of the second fundamental form of an isometric immersion of a space form with negative curvature into a space form so that the extrinsic curvature is negative. If the immersion has flat normal bundle, we prove that its second fundamental form grows exponentially.

On Algebraic-Geometric Methods for Constructing Submanifolds with Flat Normal Bundle and Holonomic Net of Curvature Lines

In this paper we propose a generalization of Krichever’s algebraic-geometric construction of orthogonal coordinate systems in a flat space. In the theory of integrable systems of hydrodynamic type a fundamental role is also played by orthogonal coordinates in some special nonflat spaces. The most important class of such spaces is given by metrics of submanifolds in flat spaces that have flat normal bundle and holonomic net of curvature lines, which defines orthogonal coordinates on the submanifold. We propose a method for constructing such submanifolds from algebraic-geometric data. Explicit examples are presented.

On algebraic-geometry methods for constructing submanifolds with flat normal bundle and holonomic net of curvature lines

В данной статье предложено обобщение алгебро-геометрической конструкции Кричевера построения ортогональных систем координат в плоском пространстве. В теории интегрируемых систем гидродинамического типа фундаментальную роль играют также ортогональные координаты в некоторых специальных неплоских пространствах. Важнейший класс таких пространств задается метриками подмногообразий в плоском пространстве с плоской нормальной связностью и голономной сетью линий кривизны, определяющей ортогональные координаты на подмногообразии. Предложена конструкция построения таких подмногообразий по алгебро-геометрическим данным. Приведены явные примеры.

Hypersurfaces with constant principal curvatures in Sn×R and Hn×R

In this paper, we classify the hypersurfaces in Sn×R and Hn×R, n≠3, with g distinct constant principal curvatures, g∈{1,2,3}, where Sn and Hn denote the sphere and hyperbolic space of dimension n, respectively. We prove that such hypersurfaces are isoparametric in those spaces. Furthermore, we find a necessary and sufficient condition for an isoparametric hypersurface in Sn×R and Hn×R with flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces Rn+2⊃Sn×R and Ln+2⊃Hn×R, having constant principal curvatures.

Variational characterizations of $$\xi $$-submanifolds in the Eulicdean space $${{\mathbb {R}}}^{m+p}$$

$$\xi $$ -submanifold in the Euclidean space $${{\mathbb {R}}}^{m+p}$$ is a natural extension of the concept of self-shrinker to the mean curvature flow in $${{\mathbb {R}}}^{m+p}$$ . It is also a generalization of the $$\lambda $$ -hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for $$\xi $$ -submanifolds are established. First, it is shown that a submanifold in $${{\mathbb {R}}}^{m+p}$$ is a $$\xi $$ -submanifold if and only if its modified mean curvature is parallel when viewed as a submanifold in the Gaussian space $$\big ({{\mathbb {R}}}^{m+p},e^{-\frac{1}{m}|x|^2}\left\langle \cdot ,\cdot \right\rangle \big )$$ ; then, two generalized weighted volume functionals $$V_\xi $$ and $${{\bar{V}}}_\xi $$ are defined and it is proved that $$\xi $$ -submanifolds can be characterized as the critical points of these two functionals; also, the corresponding second variation formulas are computed. Finally, we introduce the VP-variations and the corresponding W-stability for $$\xi $$ -submanifolds which are then systematically studied. As the main result, it is proved that m-planes are the only complete, W-stable and properly immersed $$\xi $$ -submanifolds with flat normal bundle.

On the linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

Let $$M^{n}$$ be an n-dimensional complete linear Weingarten spacelike submanifold immersed with parallel normalized mean curvature vector field and flat normal bundle in a locally symmetric semi-Riemannian space $$L_{p}^{n+p}$$ of index p, which obeys standard curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). In this setting, our purpose is to establish sufficient conditions guaranteeing that such a spacelike submanifold $$M^{n}$$ be either totally umbilical or isometric to an isoparametric hypersurface of a totally geodesic submanifold $$L_{1}^{n+1}\hookrightarrow L_{p}^{n+p}$$, with two distinct principal curvatures, one of which is simple. Our approach is based on a suitable Simons type formula jointly with a version of the Omori–Yau’s generalized maximum principle for a Cheng–Yau’s modified operator.

Conformally flat submanifolds with flat normal bundle

We prove that any conformally flat submanifold with flat normal bundle in a conformally flat Riemannian manifold is locally holonomic, that is, admits a principal coordinate system. As one of the consequences of this fact, it is shown that the Ribaucour transformation can be used to construct an associated large family of immersions with induced conformally flat metrics holonomic with respect to the same coordinate system.

LW-surfaces with higher codimension and Liebmann’s Theorem in the hyperbolic space

We deal with complete surfaces immersed with flat normal bundle and parallel normalized mean curvature vector field in the hyperbolic space $${\mathbb {H}}^{2+p}$$ . Supposing that such a surface $$M^{2}$$ satisfies a linear Weingarten condition of the type $$K=aH+b$$ for some appropriate real constants a and b, where H and K denote the mean and Gaussian curvatures, respectively, we show that $$M^{2}$$ must be either totally umbilical or isometric to one of the following flat surfaces: $${\mathbb {S}}^{1}(r)\times {\mathbb {R}}$$ , $$\mathbb S^{1}(r)\times {\mathbb {S}}^1(\sqrt{{\tilde{r}}^2-r^2})$$ or $$\mathbb S^{1}(r)\times {\mathbb {H}}^1(-\sqrt{{\tilde{r}}^2+r^2})$$ . Furthermore, we obtain a version of the classical Liebmann’s Theorem (Liebmann in Math Phys Klasse 44–55, 1899) showing that the only compact (without boundary) surfaces having positive constant Gaussian curvature, immersed with flat normal bundle in the hyperbolic space $${\mathbb {H}}^{2+p}$$ , are the totally umbilical round spheres.

The Gauss–Bonnet–Chern mass of higher-codimension graphs

We give an explicit formula for the Gauss-Bonnet-Chern mass of an asymptotically flat graphical manifold of arbitrary codimension and use it to prove the positive mass theorem and the Penrose inequality for graphs with flat normal bundle.

A Bernstein-type theorem for -submanifolds with flat normal bundle in the Euclidean spaces

$\xi$-Submanifolds in the Euclidean spaces are a natural extension of self-shrinkers and a generalization of $\lambda$-hypersurfaces. Moreover, $\xi$-submanifolds are expected to take the place of submanifolds with parallel mean curvature vector. In this paper, we establish a Bernstein-type theorem for $\xi$-submanifolds in the Euclidean spaces. More precisely, we prove that an $n$-dimensional smooth graphic $\xi$-submanifold with flat normal bundle in $\mathbb{R}^{n+p}$ is an affine $n$-plane.

Pseudo-parallel surfaces of $$\mathbb {S}_c^n \times \mathbb {R}$$ S c n × R and $$\mathbb {H}_c^n \times \mathbb {R}$$ H c n × R

In this work we give a characterization of pseudo-parallel surfaces in $$\mathbb {S}_c^n \times \mathbb {R}$$ and $$\mathbb {H}_c^n\times \mathbb {R}$$ , extending an analogous result by Asperti-Lobos-Mercuri for the pseudo-parallel case in space forms. Moreover, when $$n=3$$ , we prove that any pseudo-parallel surface has flat normal bundle. We also give examples of pseudo-parallel surfaces which are neither semi-parallel nor pseudo-parallel surfaces in a slice. Finally, when $$n\ge 4$$ we give examples of pseudo-parallel surfaces with non vanishing normal curvature.

Linear Weingarten submanifolds in the hyperbolic space

We use suitable maximum principles in order to obtain characterization results concerning n-dimensional linear Weingarten submanifolds immersed with globally flat normal bundle and parallel normalized mean curvature vector field in the $$(n+p)$$ -dimensional hyperbolic space $${\mathbb {H}}^{n+p}$$ . In particular, when $$p=2$$ , we present complete descriptions of these submanifolds.

HIGHER CODIMENSIONAL UEDA THEORY FOR A COMPACT SUBMANIFOLD WITH UNITARY FLAT NORMAL BUNDLE

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher codimensional generalization of Ueda’s result, we give a sufficient condition for the existence of a nonsingular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semipositive curvature on a nef line bundle.

Einstein submanifolds with flat normal bundle in space forms are holonomic

A well-known result asserts that any isometric immersion with flat normal bundle of a Riemannian manifold with constant sectional curvature into a space form is (at least locally) holonomic. In this note, we show that this conclusion remains valid for the larger class of Einstein manifolds. As an application, when assuming that the index of relative nullity of the immersion is a positive constant we conclude that the submanifold has the structure of a generalized cylinder over a submanifold with flat normal bundle.

Harmonic p-forms on Hadamard manifolds with finite total curvature

In the present note, the geometric structures and topological properties of harmonic p-forms on a complete noncompact submanifold \(M^{n}(n\ge 4)\) immersed in Hadamard manifold \(N^{n+m}\) are discussed, where \(M^{n}\) and \(N^{n+m}\) are assumed to have flat normal bundle and pure curvature tensor, respectively. Firstly, under the assumption that \(M^{n}\) satisfies the \((\mathcal {P}_\rho )\) property (i.e., the weighted Poincare inequality holds on \(M^{n}\)) and the \((p,n-p)\)-curvature of \(N^{n+m}\) is not less than a given negative constant, using Moser iteration, the space of all \(L^{2}\) harmonic p-forms on \(M^{n}\) is proven to have finite dimensions if \(M^{n}\) has finite total curvature. Furthermore, if the total curvature is small enough or \(M^{n}\) has at most Euclidean volume growth, two vanishing theorems are, respectively, established for harmonic p-forms. Note that the two vanishing theorems extend several previous results obtained by H. Z. Lin.