Extrinsic Sphere Research Articles

Weakly Einstein Equivalence in a Golden Space Form and Certain CR-submanifolds

ABSTRACT It is known that the Golden space form may not be an Einstein manifold. In this paper, it is shown that the condition to be Einstein for a Golden space form is equivalent to being weakly Einstein. In addition, the partial geodesic and cyclic parallelism of the CR-submanifolds of a Golden space are examined, and the case of the constant Golden sectional curvature is determined. Moreover, the CR-submanifolds with semi-flat normal connection are studied and an inequality is obtained. The equality case of this inequality is also checked. We also consider the totally umbilical CR-submanifold of Golden Riemannian manifolds and show that such submanifolds are totally geodesic under certain conditions. Furthermore, we obtain an inequality involving the scalar curvature of CR-submanifold and check the existence of extrinsic spheres in Golden space forms.

Some differentiable sphere theorems

In this paper, we obtain several new intrinsic and extrinsic differentiable sphere theorems via Ricci flow. For intrinsic case, we show that a closed simply connected $$n(\ge 4)$$ -dimensional Riemannian manifold M is diffeomorphic to $$\mathbb {S}^n$$ if one of the following conditions holds pointwisely: $$\begin{aligned} (i)\ R_0>\left( 1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right) K_{max};\quad \ (ii)\ \frac{Ric^{[4]}}{4(n-1)}>\left( 1-\frac{6(\sqrt{10}-3)}{n-1}\right) K_{max}. \end{aligned}$$ Here $$K_{max}$$ , $$Ric^{[k]}$$ and $$R_0$$ stand for the maximal sectional curvature, the k-th weak Ricci curvature and the normalized scalar curvature. For extrinsic case, i.e., when M is a closed simply connected $$n(\ge 4)$$ -dimensional submanifold immersed in $$\bar{M}$$ . We prove that M is diffeomorphic to $$\mathbb {S}^n$$ if it satisfies some curvature pinching conditions. The only involved extrinsic quantities in our pinching conditions are the maximal sectional curvature $$\bar{K}_{max}$$ and the squared norm of mean curvature vector $$\left|H\right|^2$$ . More precisely, we show that M is diffeomorphic to $$\mathbb {S}^n$$ if one of the following conditions holds: It is worth pointing out that, in the proof of extrinsic case, we apply suitable complex orthonormal frame and simplify the calculations considerably. We also emphasize that both of the pinching constants in (2) and (3) are optimal for $$n=4$$ .

CIRCLES ALONG A RIEMANNIAN MAP AND CLAIRAUT RIEMANNIAN MAPS

We first extend Yano-Nomizu's theorem, which characterizes extrinsic spheres in a Riemannian manifold, for Riemannian maps. Then we introduce Clairaut Riemannian maps, give an example and obtain necessary and sufficient conditions for a Riemannian map to be Clairaut type.

On submersions of the complex indicatrix

In this paper we study the complex indicatrix associated to a complex Finsler space as an embedded CR - hypersurface of the holomorphic tangent bundle, considered in a fixed point. Following the study of CR - submanifolds of a K?hler manifold, there are investigated some properties of the complex indicatrix as a real submanifold of codimension one, using the submanifold formulae and the fundamental equations. As a result, the complex indicatrix is an extrinsic sphere of the holomorphic tangent space in each fibre of a complex Finsler bundle. Also, submersions from the complex indicatrix onto an almost Hermitian manifold and some properties that can occur on them are studied. As application, an explicit submersion onto the complex projective space is provided.

Totally umbilical hemislant submanifolds of LP-Sasakian manifold

This paper is summarized as follows. In the first section we have given a brief history about slant and hemi-slant submanifold of LP-Sasakian manifold. This section is followed by some preliminaries about LP-Sasakian manifold. Finally, we have derived some interesting results on the existence of extrinsic sphere for totally umbilical hemi-slant submanifold of LP-Sasakian manifold.

On lightlike geometry in indefinite Kenmotsu manifolds

Abstract We investigate some geometric aspects of lightlike hypersurfaces of indefinite Kenmotsu manifolds, tangent to the structure vector field, by paying attention to the geometry of leaves of integrable distributions. Theorems on parallel vector fields, Killing distribution, geodesibility of their leaves are obtained. The geometric configuration of such lightlike hypersurfaces and leaves of its screen integrable distributions are established. We show that no totally contact umbilical leaf of a screen integrable distribution of a lightlike hypersurface can be an extrinsic sphere. We also prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle.

CHARACTERIZATIONS OF SOME ISOMETRIC IMMERSIONS IN TERMS OF CERTAIN FRENET CURVES

We give criterions for a submanifold to be an extrinsic sphere and to be a totally geodesic submanifold by observing some Frenet curves of order 2 on the submanifold. We also characterize constant isotropic im- mersions into arbitrary Riemannian manifolds in terms of Frenet curves of proper order 2 on submanifolds. As an application we obtain a char- acterization of Veronese embeddings of complex projective spaces into complex projective spaces.

Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motions in the extrinsic balls, i.e. lower bounds for the time it takes (on average) for Brownian particles to diffuse within the extrinsic ball from a given starting point before they hit the boundary of the extrinsic ball. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above.

Screen Integrable Lightlike Hypersurfaces of Indefinite Sasakian Manifolds

We investigate lightlike hypersurfaces of indefinite Sasakian manifolds, tangent to the structure vector field ξ and whose screen distribution is integrable. We prove some results on parallel vector fields and on a leaf of the integrable distribution \(D_{0} \perp \langle\xi\rangle\) of this class. A theorem on a geometrical configuration of the screen distribution is obtained. We show that any totally contact umbilical leaf of a screen integrable distribution of a lightlike hypersurface is an extrinsic sphere.

Indefinite Extrinsic Spheres

We study the lightlike geometry of the second fundamental form of the intersection between an algebraic hypersurface in $\mathbb{C}_{s}^{n}$ and a pseudosphere $S_{2s}^{2n-1}(2/\sqrt{k})$ including a class of extrinsic spheres which are not homotopy spheres.

Extrinsic spheres in a real space form

Let M be an n-dimensional orientable compact hypersurface in an (n+1)-dimensional real space form M(c), n ≥2. If the lengths ∥ℝ∥, ∥double-struck A∥ and ∥∇α∥ of the curvature tensor field R, the shape operator A, the gradient ∇α of the mean curvature α and the scalar curvature S of the hypersurface M satisfy the inequality (Equation Presented) where δ = min Ric = min (Equation Presented) Ricp(v), Ric is Ricci curvature of the hypersurface, then it is shown that M is an extrinsic sphere in M(c). In particular we deduce that the condition 1/2 ∥R∥2 ≤ δ ∥A∥2 - n(n - 1) ∥∇α∥2 characterizes spheres in the Euclidean space Rn+l among the compact orientable hypersurfaces whose Ricci curvatures are bounded below by a constant δ > 0.

Totally Umbilic Isometric Immersions and Curves of Order Two

In this paper we study submanifolds by use of extrinsic shapes of some curves having points of proper order 2, and give a condition that they are totally umbilic. This gives an extension of Nomizu-Yano’s result in [7] on a characterization of extrinsic spheres.

Mean curvature of extrinsic spheres in submanifolds of real space forms

Given a submanifold P m with the Hilbert-Schmidt norm of its second fundamental form bounded from above, in a real space form of constant curvature % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaacY % catuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-Pi8 % lnaaCaaaleqabaGaamOBaaaakiaacIcacaWGIbGaaiykaiaacYcaaa % a!478C! $$b,\mathbb{K}^n (b),$$ we have obtained a lower bound for the norm of the mean curvature normal vector field of extrinsic spheres with sufficiently small radius in P m in terms of the mean curvature of the geodesic spheres in % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFkc-sdaahaaWcbeqa % aiaad2gaaaGccaGGOaGaamOyaiaacMcacaGGSaaaaa!45F4! $$\mathbb{K}^m (b),$$ with same radius, and the mean curvature of P m .

On the existence of spherically bent submanifolds, an analogue of a theorem of E. Cartan

In this article an analogue of E. Cartan's theorem about the existence of (local) totally geodesic submanifolds with a prescribed tangent plane is proved, namely for the existence of spherically bent submanifolds (= extrinsic spheres); also a global version is deduced.

A characterization of extrinsic spheres in a Riemannian manifold

We give a characterization of a totally umbilic submanifold $M^n$ with parallel mean curvature vector of a Riemannian manifold $\tilde{M}^{n+p}$, that is an extrinsic sphere $M^n$ of $\tilde{M}^{n+p}$, in terms of the extrinsic shape of circles on $M^n$ in the ambient manifold $\tilde{M}^{n+p}$. This characterization is an improvement of Nomizu and Yano's result ([2]).

On a class of submanifolds carrying an extrinsic totally umbilical foliation

In this paper we give a conformal classification of all euclidean submanifolds carrying a parallel principal curvature normal under the intrinsic additional assumption that the associated conformal conullity is involutive and the leaves are extrinsic spheres in the submanifold in the sense of Nomizu. We also provide several applications of this result.

Mean Curvature and Volume of Extrinsic Spheres in Hypersurfaces of Real Space Forms

Given a hypersurface Pn-1 in a real space form of constantcurvature b, \(\mathbb{K}^n \left( b \right)\), we have obtained a lower bound for the norm of the mean curvature normal vector field of extrinsicspheres in Pn-1 in terms of the mean curvature of the geodesic spheres in \(\mathbb{K}^{n - 1} \left( b \right)\), with the same radius, and the meancurvature of Pn-1, characterizing too the equality.

On totally umbilical QR-submanifolds of quaternion Kaehlerian manifolds

We introduce the notion of generalised 3-Sasakian structure on a manifold and show that a totally umbilical, but not totally geodesic, proper QR-submanifold of a quaternion Kaehlerian manifold is an extrinsic sphere and inherits such a structure.

On proper helices and extrinsic spheres in pseudo-Riemannian geometry

In this paper, we define the notion of a proper helix of order $d$ in a pseudo-Riemannian manifold and investigate those curves in a totally umbilical pseudo-Riemannian submanifold.

Totally umbilical manifolds inG(2,n). I

It is well known that every totally umbilical submanifold of a space of constant curvature is either a small sphere or is totally geodesic. B.-Y. Chen has classified totally umbilical submanifolds of compact, rank one, symmetric spaces ([4], [5]): in particular, they are all extrinsic spheres, that is, they have a parallel mean curvature vector H (or are totally geodesic). In this paper totally umbilical submanifolds Fl of dimension l ≥ 3 are classified in the irreducible symmetric space that is "next in complexity": Grassmann manifold G(2, n). Such submanifolds are either 1) totally geodesic [3] or 2) extrinsic spheres [small spheres in totally geodesic spheres; their position in G(2, n) is described here] or 3) essentially totally umbilical (H ≠ 0, ∇ ⊥ H ≠ 0). If the submanifold is of type 3), then it is either a) an umbilical hypersurface of nonconstant mean curvature in totally geodesic S1 × S1 ⊂ G(2, n) or b) an "oblique diagonal," a diagonal of the product of two small spheres of different radii in totally geodesic Sl+1 × Sl+1 ⊂ G(2, n) (it has constant mean and sectional curvatures). Submanifolds 3a) and 3b) are described completely. The latter of the two negates two of Chen's conjectures. It is shown that submanifold Fl ⊂ El+2 (l ≥ 3) with a totally umbilical Grassmannian image has a totally geodesic Grassmannian image and is classifiable [11].