Flat Normal Bundle Research Articles

Einstein submanifolds with parallel mean curvature

We provide a classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature. This extends a previous result due to Dajczer and Tojeiro (Tohoku Math J (2) 45:43–49, 1993) for isometric immersions of Riemannian manifolds with constant sectional curvature.

On the curvature of Einstein–Hermitian surfaces

We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal bundle. We also give another proof of the fact that a compact complex surface together with an Einstein–Hermitian metric of positive orthogonal bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini–Study metric up to rescaling. This result relaxes the Kähler condition in Berger’s theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.

On biconservative surfaces in 4-dimensional Euclidean space

In this paper, we study non-CMC biconservative surfaces with flat normal bundle in the Euclidean 4-space E4. First, we determine locally all parallel normalized mean curvature vector (PNMCV) biconservative surfaces in E4, showing that they are certain rotational surfaces. Then, we consider meridian surfaces lying on a rotational hypersurface of E4 and construct a family of biconservative surfaces with flat normal bundle and non-parallel normalized mean curvature vector.

P-Harmonic l-forms on Complete Noncompact Submanifolds in Sphere with Flat Normal Bundle

In this paper, we investigate a complete noncompact submanifold $$M^m$$ in a sphere $$S^{m+t}$$ with flat normal bundle. We prove that the dimension of the space of $$L^p$$ p-harmonic l-forms (when $$m\ge 4$$ , $$2\le l\le m-2$$ and when $$m=3$$ , $$l=2$$ ) on M is finite if the total curvature of M is finite and $$m\ge 3$$ . We also obtain that there are no nontrivial $$L^p$$ p-harmonic l-forms on M if the total curvature is bounded from above by a constant depending only on m, p, l.

P-harmonic [formula omitted]-forms on Riemannian manifolds with a weighted Poincaré inequality

Given a Riemannian manifold with a weighted Poincaré inequality, in this paper, we will show some vanishing type theorems for p-harmonic ℓ-forms on such a manifold. We also prove a vanishing result on submanifolds in Euclidean space with flat normal bundle. Our results can be considered as generalizations of the work of Lam, Li–Wang, Lin, and Vieira (see Lam (2008), Li and Wang (2001), Lin (2015), Vieira (2016)). Moreover, we also prove a vanishing and splitting type theorem for p-harmonic functions on manifolds with Spin (9) holonomy provided a (p,p,λ)-Sobolev type inequality which can be considered as a general Poincaré inequality holds true.

Conformal geometry of marginally trapped surfaces in $$\mathbb {S}^4_1$$ S 1 4

A spacelike surface S immersed in \(\mathbb {S}^4_1\) is marginally trapped if its mean curvature vector is everywhere lightlike. On any oriented spacelike surface S immersed in \(\mathbb {S}^4_1\) we show that a choice of orientation of the normal bundle \(\nu (S)\) determines a smooth map \(G: S \rightarrow \mathbb {S}^3\) which we call the null Gauss map of S. If S is marginally trapped we show that G is a conformal immersion away from the zeros of certain quadratic Hopf-differential of S and so the surface G(S) is uniquely determined up to conformal transformations of \(\mathbb {S}^3\) by two invariants: the normal Hopf differential \(\kappa \) and the schwartzian derivative s. These invariants plus an additional quadratic differential \(\delta \) are related by a differential equation and determine the geometry of S up to ambient isometries of \(\mathbb {S}^4_1\). This allows us to obtain a characterization of marginally trapped surfaces S whose null Gauss image is a constrained Willmore surface in \(\mathbb {S}^3\) in the sense of Bohle et al. (Calc Var Partial Differ Equ 32:263–277, 2008). As an application of these results we construct and study integrable non-trivial one-parameter deformations of marginally trapped surfaces with non-zero parallel mean curvature vector and those with flat normal bundle.

Spinor representation of Lorentzian surfaces in R2,2

We prove that an isometric immersion of a simply connected Lorentzian surface in R 2 , 2 is equivalent to a normalised spinor field solution of a Dirac equation on the surface. Using the quaternions and the Lorentz numbers, we also obtain an explicit representation formula of the immersion in terms of the spinor field. We then apply the representation formula in R 2 , 2 to give a new spinor representation formula for Lorentzian surfaces in 3-dimensional Minkowski space. Finally, we apply the representation formula to the local description of the flat Lorentzian surfaces with flat normal bundle and regular Gauss map in R 2 , 2 , and show that these surfaces locally depend on four real functions of one real variable, or on one holomorphic function together with two real functions of one real variable, depending on the sign of a natural invariant.

ON SPACELIKE ROTATIONAL SURFACES WITH POINTWISE 1-TYPE GAUSS MAP

In this paper, we study a class of spacelike rotational surfaces in the Minkowski 4-space <TEX>$\mathbb{E}^4_1$</TEX> with meridian curves lying in 2-dimensional spacelike planes and having pointwise 1-type Gauss map. We obtain all such surfaces with pointwise 1-type Gauss map of the first kind. Then we prove that the spacelike rotational surface with flat normal bundle and pointwise 1-type Gauss map of the second kind is an open part of a spacelike 2-plane in <TEX>$\mathbb{E}^4_1$</TEX>.

Bernstein-type theorem of translating solitons in arbitrary codimension with flat normal bundle

We give a proof for a Bernstein-type theorem of complete graphs of translating solitons in higher codimension. In the case of hypersurfaces, Bao–Shi showed that a translating soliton whose image of the Gauss map is contained in a compact subset in an open hemisphere is a hyperplane. This means that there is no nontrivial translating soliton whose slope is bounded. In the present article, we generalize this theorem in arbitrary codimension. Moreover we obtain an optimal growth condition which allows unbounded slopes. As a corollary, our result covers a classical Bernstein-type theorem for minimal submanifolds.

On the Structure of Submanifolds in Euclidean Space with Flat Normal Bundle

In this paper we study the structure of an immersed submanifold Mn in a Riemannian manifold with flat normal bundle in two ways. Firstly, we prove that if Mn is compact and satisfies some pointwise pinching condition, and assume further that the ambient space has pure curvature tensor and non-negative isotropic curvature, then the Betti numbers βp(M) = 0 for 2 ≤ p ≤ n−2. Secondly, suppose that Mn is a complete non-compact submanifold in the Euclidean space with finite total curvature in the sense that its traceless second fundament form has finite Ln-norm, then we show that the spaces of L2 harmonic p-forms on Mn have finite dimensions for all 2 ≤ p ≤ n−2.

Vanishing theorems for [formula omitted] harmonic forms on complete submanifolds in Euclidean space

The aim of this paper is to prove vanishing theorems for L2 harmonic forms of higher order on complete non-compact submanifolds in Euclidean space. Firstly, by assuming that the submanifold has flat normal bundle, we can explicitly express the Weitzenböck formulae for harmonic p-forms. Using this formulae, we can obtain some L2 vanishing theorems by adding a relation of the square length of the second fundamental form with the squared mean curvature, or by assuming that the total curvature of the submanifold is bounded from above by an explicit positive constant. Secondly, by using a monotonicity formulae for general harmonic forms, we obtain a vanishing theorem under an appropriate decay of the norm of the second fundamental form.

The positive mass theorem and Penrose inequality for graphical manifolds

We give, via elementary methods, explicit formulas for the ADM mass which allow us to conclude the positive mass theorem and Penrose inequality for a class of graphical manifolds which includes, for instance, those with flat normal bundle.

Complete noncompact submanifolds with flat normal bundle

Complete noncompact submanifolds with flat normal bundle

Pseudo-Parallel Legendrian Submanifolds With Flat Normal Bundle of Sasakian Space Forms

Let M n be a Legendrian submanifold with flat normal bundle of a Sasakian space form 2 n +1( c ). Further, M n is said to be pseudo-parallel if its second fundamental form h satisfies R ( X, Y ) · h = L ( X ∧ Y · h ) . In this article we shall prove that M is semi-parallel or totally geodesic and if M satisfies L then it is minimal in case of n ≥ 2. Moreover, we show that if M n is also a H-umbilical submanifold then either M n is L = , or n = 1.

Ricci generalized pseudo-parallel Legendrian submanifolds with flat normal bundle of Sasakian space forms

Ricci generalized pseudo-parallel Legendrian submanifolds with flat normal bundle of Sasakian space forms

The Gauss–Bonnet–Chern mass for graphic manifolds

In this paper, we prove a positive mass theorem and Penrose-type inequality of the Gauss–Bonnet–Chern mass $$m_2$$ for the graphic manifold with flat normal bundle.

On the spinorial representation of spacelike surfaces into 4-dimensional Minkowski space

We prove that an isometric immersion of a simply connected Riemannian surface M in four-dimensional Minkowski space, with given normal bundle E and given mean curvature vector H→∈Γ(E), is equivalent to a normalized spinor field φ∈Γ(ΣE⊗ΣM) solution of a Dirac equation Dφ=H→⋅φ on the surface. Using the immersion of the Minkowski space into the complex quaternions, we also obtain a representation of the immersion in terms of the spinor field. We then use these results to describe the flat spacelike surfaces with flat normal bundle and regular Gauss map in four-dimensional Minkowski space, and also the flat surfaces in three-dimensional hyperbolic space, giving spinorial proofs of results by J.A. Gálvez, A. Martínez and F. Milán.

On Willmore surfaces in 𝑆ⁿ of flat normal bundle

We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in S n S^n must be located in some S 3 ⊂ S n S^3\subset S^n , from which we characterize the Clifford torus as the only non-equatorial homogeneous minimal surface in S n S^n with flat normal bundle, which improves a result of K. Yang. Then we derive that every Willmore two sphere with flat normal bundle in S n S^n is conformal to a minimal surface with embedded planer ends in R 3 \mathbb {R}^3 . We also point out that for a class of Willmore tori, they have a flat normal bundle if and only if they are located in some S 3 S^3 . In the end, we show that a Willmore surface with flat normal bundle must locate in some S 6 S^6 .

Semiumbilic points for minimal surfaces in Euclidean $$4$$ 4 -space

For a minimal surface in the Euclidean 4-space, a semiumbilic point is at which its normal curvature vanishes. We prove that the semiumbilic points are isolated, if the normal curvature neither changes its sign nor vanishes identically. We also show that any entire minimal surface with flat normal bundle is an affine plane.

On the Components of the Push-out Space with Certain Indices

Given an immersion of a connected, m -dimensional manifold M without boundary into the Euclidean (m+k) -dimensional space, the idea of the push-out space of the immersion under the assumption that immersion has flat normal bundle is introduced in [3]. It is known that the push-out space has finitely many path-connected components and each path-connected component can be assigned an integer called the index of the component. In this study, when M is compact, we give some new results on the push-out space. Especially it is proved that if the push-out space has a component with index 1 , then the Euler number of M is 0 and if the immersion has a co-dimension 2 , then the number of path-connected components of the push-out space with index (m-1) is at most 2.