scholarly journals ON SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM

2009 ◽  
Vol 51 (3) ◽  
pp. 669-680 ◽  
Author(s):  
G. PACELLI BESSA ◽  
M. SILVANA COSTA
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
Complete Riemannian Manifold ◽  
Hadamard Manifold ◽  
Fundamental Tone ◽  
Finite Topology ◽  
Bounded Below

AbstractBased on the ideas of Bessa, Jorge and Montenegro (Comm. Anal. Geom., vol. 15, no. 4, 2007, pp. 725–732) we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 is proper (compact if N is compact). In addition, if N is Hadamard, then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realised as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.

scholarly journals A characterization of complete Riemannian manifolds minimally immersed in the unit sphere

1993 ◽  
Vol 131 ◽  
pp. 127-133 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Clifford Torus ◽  
Veronese Surface ◽  
The Second Fundamental Form ◽  
Complete Riemannian Manifolds

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totallygeodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

Immersions with Semi-Definite Second Fundamental Forms

1975 ◽  
Vol 27 (3) ◽  
pp. 610-617 ◽  
Author(s):  
Leo B. Jonker
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Fundamental Form ◽  
Isometric Immersion ◽  
Tangent Space ◽  
Normal Component ◽  
Second Fundamental Form ◽  
Normal Space ◽  
The Second Fundamental Form ◽  
Image Position

Let M be a. complete connected Riemannian manifold of dimension n and let £:M → Rn+k be an isometric immersion into the Euclidean space Rn+k. Let ∇ be the connection on Mn and let be the Euclidean connection on Rn+k. Also letdenote the second fundamental form B(X, Y) = (xY)→. Here TP(M) denotes the tangent space at p, NP(M) the normal space and (…)→ the normal component.

scholarly journals PINCHING THEOREMS FOR A COMPACT MINIMAL SUBMANIFOLD IN A COMPLEX PROJECTIVE SPACE

2008 ◽  
Vol 77 (1) ◽  
pp. 99-114
Author(s):  
MAYUKO KON
Keyword(s):  
Projective Space ◽  
Sectional Curvature ◽  
Fundamental Form ◽  
Additional Condition ◽  
Complex Projective Space ◽  
Second Fundamental Form ◽  
Minimal Submanifold ◽  
Normal Connection ◽  
The Second Fundamental Form ◽  
Complex Projective

AbstractWe give a formula for the Laplacian of the second fundamental form of an n-dimensional compact minimal submanifold M in a complex projective space CPm. As an application of this formula, we prove that M is a geodesic minimal hypersphere in CPm if the sectional curvature satisfies K≥1/n, if the normal connection is flat, and if M satisfies an additional condition which is automatically satisfied when M is a CR submanifold. We also prove that M is the complex projective space CPn/2 if K≥3/n, and if the normal connection of M is semi-flat.

scholarly journals PSEUDO-PARALLEL KAEHLERIAN SUBMANIFOLDS IN COMPLEX SPACE FORMS

2020 ◽  
pp. 321
Author(s):  
Ahmet Yildiz
Keyword(s):  
Sectional Curvature ◽  
Fundamental Form ◽  
Complex Space ◽  
Space Form ◽  
Dimensional Space ◽  
Second Fundamental Form ◽  
Holomorphic Sectional Curvature ◽  
Space Forms ◽  
Totally Geodesic ◽  
Complex Space Forms

Let $\tilde{M}^{m}(c)$ be a complex $m$-dimensional space form of holomorphic sectional curvature $c$ and $M^{n}$ be a complex $n$-dimensional Kaehlerian submanifold of $\tilde{M}^{m}(c).$ We prove that if $M^{n}$ is pseudo-parallel and $Ln-\frac{1}{2}(n+2)c\geqslant 0$ then $M$ $^{n}$ is totally geodesic. Also, we study Kaehlerian submanifolds of complex space form with recurrent second fundamental form.

scholarly journals Energy-minimizing maps from manifolds with nonnegative Ricci curvature

2019 ◽  
pp. 1950083 ◽  
Author(s):  
James Dibble
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Universal Covering ◽  
Complete Riemannian Manifold ◽  
Conjugate Points ◽  
Nonnegative Ricci Curvature ◽  
Universal Covering Space ◽  
Asymptotic Geometry ◽  
Bounded Below ◽  
Minimizing Maps

The energy of any [Formula: see text] representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only if the original map is totally geodesic. This conclusion also holds under the weaker assumption that the domain is finitely covered by a diffeomorphic product, and its universal covering space splits isometrically as a product with a flat factor, in a commutative diagram that follows from the Cheeger–Gromoll splitting theorem.

On the Geometry of Submersions

2021 ◽  
pp. 199-208
Author(s):  
Abdigappar Narmanov ◽  
Xurshid Sharipov
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Submersion ◽  
Complete Riemannian Manifold ◽  
Complete Manifold ◽  
Geodesic Foliation ◽  
Metric Function

Subject of present paper is the geometry of foliation defined by submersions on complete Riemannian manifold. It is proven foliation defined by Riemannian submersion on the complete manifold of zero sectional curvature is total geodesic foliation with isometric leaves. Also it is shown level surfaces of metric function are conformally equivalent.

scholarly journals Higher-order geometric flow of hypersurfaces in a Riemannian manifold

2019 ◽  
Vol 30 (13) ◽  
pp. 1940005
Author(s):  
Zonglin Jia ◽  
Youde Wang
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Sectional Curvature ◽  
Present Situation ◽  
Higher Order ◽  
Complete Riemannian Manifold ◽  
High Order ◽  
Geometric Flows ◽  
Integer Part ◽  
Geometric Flow

In this paper, we consider the high-order geometric flows of a compact submanifolds [Formula: see text] in a complete Riemannian manifold [Formula: see text] with [Formula: see text], which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on [Formula: see text]. Precisely, we show that if [Formula: see text] is strictly larger than the integer part of [Formula: see text] and [Formula: see text] is an immersion for all [Formula: see text] and if [Formula: see text] is bounded by a constant which relies on the injectivity radius [Formula: see text] and sectional curvature [Formula: see text] of [Formula: see text], then [Formula: see text] must be [Formula: see text].

scholarly journals GRADIENT AND HESSIAN ESTIMATES FOR DIRICHLET AND NEUMANN EIGENFUNCTIONS

2020 ◽  
Vol 71 (1) ◽  
pp. 379-394
Author(s):  
Feng-Yu Wang
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Integral Formulas ◽  
Manifold With Boundary ◽  
Hessian Estimates

Abstract We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.

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