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1.
We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres with parallel mean curvature vector field and parallel Weingarten operator associated to the mean curvature vector field. 相似文献
2.
In this paper we proved a better estimate as well as generalized to higher codimensions of a theorem of Y.B. Shen on complete submanifolds with parallel mean curvature vector in a hyperbolic space. 相似文献
3.
Let M n be an n-dimensional closed submanifold of a sphere with parallel normalized mean curvature vector. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M n , respectively. Assume that the fundamental group \({\pi_{1}(M^{n})}\) of M n is infinite and \({S\, \leqslant\, S(H)=n+\frac{n^{3}H^{2}}{2(n-1)}-\frac{n(n-2)H}{2(n-1)}\sqrt{n^{2}H^{2}+4(n-1)}}\), then S is constant, S = S(H), and M n is isometric to a Clifford torus \({S^{1}(\sqrt{1-r^{2}})\times S^{n-1}(r)}\) with \({r^{2}\leqslant \frac{n-1}{n}}\). 相似文献
4.
Our aim in this article is to study the geometry of n-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space \({\mathbb{R}^{n+p}_{q}}\) of index q, with \({1\leq q\leq p}\). Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of \({\mathbb{R}^{n+p}_{q}}\) be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in \({\mathbb{R}^{n+p}_{p}}\) and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz–Minkowski space \({\mathbb{R}^{n+1}_{1}}\). 相似文献
5.
Dr. Helmut Reckziegel 《manuscripta mathematica》1974,13(1):69-71
In a recent paper [2] K. Nomizu has shown that a natural analogue of an n-sphere in an arbitrary Riemannian manifold is an n-dimensional umbilical submanifold with non-zero parallel mean curvature vector, which he calls extrinsic sphere sometimes. This note is concerned with the question whether extrinsic spheres have a special topological or differentiable feature. 相似文献
6.
Complete space-like submanifolds in a de Sitter space with parallel mean curvature vector 总被引:6,自引:0,他引:6
Qing-ming Cheng 《Mathematische Zeitschrift》1991,206(1):333-339
7.
We study submanifolds in the Euclidean space whose mean curvature vector field is a Jacobi field. First, we characterize them and produce non-trivial (non-minimal) examples and then, we look for additional conditions which imply minimality.Research partially supported by a DGICYT grant No PB94-0705-C02-01 and by a grant of Gobierno Vasco PI95/95 相似文献
8.
Isabel M. C. Salavessa 《Bulletin of the Brazilian Mathematical Society》2010,41(4):495-530
On a Riemannian manifold $
\bar M^{m + n}
$
\bar M^{m + n}
with an (m + 1)-calibration Ω, we prove that an m-submanifold M with constant mean curvature H and calibrated extended tangent space ℝH ⋇ TM is a critical point of the area functional for variations that preserve the enclosed Ω-volume. This recovers the case described
by Barbosa, do Carmo and Eschenburg, when n = 1 and Ω is the volume element of $
\bar M
$
\bar M
. To the second variation we associate an Ω-Jacobi operator and define Ω-stability. Under natural conditions, we show that
the Euclidean m-spheres are the unique Ω-stable submanifolds of ℝ
m+n
. We study the Ω-stability of geodesic m-spheres of a fibred space form M
m+n
with totally geodesic (m + 1)-dimensional fibres. 相似文献
9.
Bang-Yen Chen 《Monatshefte für Mathematik》1980,90(3):185-194
A surfaceM in a Riemannian manifold is said to have parallel normalized mean curvature vector if the mean curvature vector is nonzero and the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. In this paper, it is proved that every analytic surface in a euclideanm-spaceE
m
with parallel normalized mean curvature vector must either lies in aE
4 or lies in a hypersphere ofE
m
as a minimal surface. Moreover, it is proved that if a Riemann sphere inE
m
has parallel normalized mean curvature vector, then it lies either in aE
3 or in a hypersphere ofE
m
as a minimal surfaces. Applications to the classification of surfaces with constant Gauss curvature and with parallel normalized mean curvature vector are also given. 相似文献
10.
11.
Leng Yan Xu Hongwei 《高校应用数学学报(英文版)》2007,22(2):153-162
A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n p)-dimensional manifold Nn p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H > 1 there exists a negative number τ(n,p, H) ∈ (-1, 0) with the property that if the sectional curvature of N is pinched in [-1, τ(n,p, H)], and if the squared length of the second fundamental form is in a certain interval, then Nn p is isometric to the hyperbolic space Hn p(-1). As a consequence, this submanifold M is congruent to Sn(1/ H2-1) or theVeronese surface in S4(1/√H2-1). 相似文献
12.
13.
Maxwell Mariano 《Monatshefte für Mathematik》2012,166(1):107-120
In this work we use a Simon??s type inequality for a suitable tensor and apply it to obtain sharp estimates for the supremum of the scalar curvature for complete spacelike submanifold M n with parallel mean curvature vector in an indefinite space form ${N^{n+p}_p(c)}$ . 相似文献
14.
Hong-Wei Xu 《Archiv der Mathematik》1993,61(5):489-496
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19.
Chuanxi Wu 《Archiv der Mathematik》1998,70(6):499-504
20.
Uğur Dursun 《Journal of Geometry》2007,86(1-2):73-80
Let M be a 3-dimensional submanifold of the Euclidean space E5 such that M is not of 1-type. We show that if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector then the normal bundle is
flat. We also prove that M is an open portion of a 3-dimensional helical cylinder if and only if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector. 相似文献