scholarly journals Null 2-type Chen surfaces

1995 ◽  
Vol 37 (2) ◽  
pp. 233-242 ◽  
Author(s):  
Shi-Jie Li
Keyword(s):  
Mean Curvature ◽  
Spectral Decomposition ◽  
Harmonic Functions ◽  
Curvature Vector ◽  
Minimal Submanifolds ◽  
Dimensional Euclidean Space ◽  
Position Vector ◽  
Mean Curvature Vector ◽  
The Mean ◽  
Image Position

Let M be an n-dimensional connected submanifold in an mdimensional Euclidean space Em. Denote by δ the Laplacian of M associated with the induced metric. Then the position vector x and the mean curvature vector H of Min Em satisfyThis yields the following fact: a submanifold M in Em is minimal if and only if all coordinate functions of Em, restricted to M, are harmonic functions. In other words, minimal submanifolds in Emare constructed from eigenfunctions of δ with one eigenvalue 0. By using (1. 1), T. Takahashi proved that minimal submanifolds of a hypersphere of Em are constructed from eigenfunctions of δ with one eigenvalue δ (≠0). In [3, 4], Chen initiated the study of submanifolds in Em which are constructed from harmonic functions and eigenfunctions of δ with a nonzero eigenvalue. The position vector x of such a submanifold admits the following simple spectral decomposition:for some non-constant maps x0and xq, where A is a nonzero constant. He simply calls such a submanifold a submanifold of null 2-type.

scholarly journals On totally umbilicalCR-submanifolds of a Kaehler manifold

1993 ◽  
Vol 16 (2) ◽  
pp. 405-408
Author(s):  
M. A. Bashir
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Lens Space ◽  
Holomorphic Sectional Curvature ◽  
Mean Curvature Vector ◽  
Kaehler Manifold ◽  
Totally Umbilical ◽  
The Mean

LetMbe a compact3-dimensional totally umbilicalCR-submanifold of a Kaehler manifold of positive holomorphic sectional curvature. We prove that if the length of the mean curvature vector ofMdoes not vanish, thenMis either diffeomorphic toS3orRP3or a lens spaceLp,q3.

scholarly journals Umbilical points on surfaces in RN

1985 ◽  
Vol 100 ◽  
pp. 135-143 ◽  
Author(s):  
Kazuyuki Enomoto
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Normal Connection ◽  
Round Sphere ◽  
Positive Gaussian Curvature ◽  
Connected Surface ◽  
Umbilical Points ◽  
The Mean

Let ϕ: M → RN be an isometric imbedding of a compact, connected surface M into a Euclidean space RN. ψ is said to be umbilical at a point p of M if all principal curvatures are equal for any normal direction. It is known that if the Euler characteristic of M is not zero and N = 3, then ψ is umbilical at some point on M. In this paper we study umbilical points of surfaces of higher codimension. In Theorem 1, we show that if M is homeomorphic to either a 2-sphere or a 2-dimensional projective space and if the normal connection of ψ is flat, then ψ is umbilical at some point on M. In Section 2, we consider a surface M whose Gaussian curvature is positive constant. If the surface is compact and N = 3, Liebmann’s theorem says that it must be a round sphere. However, if N ≥ 4, the surface is not rigid: For any isometric imbedding Φ of R3 into R4 Φ(S2(r)) is a compact surface of constant positive Gaussian curvature 1/r2. We use Theorem 1 to show that if the normal connection of ψ is flat and the length of the mean curvature vector of ψ is constant, then ψ(M) is a round sphere in some R3 ⊂ RN. When N = 4, our conditions on ψ is satisfied if the mean curvature vector is parallel with respect to the normal connection. Our theorem fails if the surface is not compact, while the corresponding theorem holds locally for a surface with parallel mean curvature vector (See Remark (i) in Section 3).

scholarly journals Pseudo-Reimannian manifolds endowed with an almost paraf-structure

1985 ◽  
Vol 8 (2) ◽  
pp. 257-266 ◽  
Author(s):  
Vladislav V. Goldberg ◽  
Radu Rosca
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Mean Curvature ◽  
Integral Manifold ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Minimal Hypersurfaces ◽  
Reimannian Manifolds ◽  
The Mean ◽  
Orthogonal Distribution

LetM˜(U,Ω˜,η˜,ξ,g˜)be a pseudo-Riemannian manifold of signature(n+1,n). One defines onM˜an almost cosymplectic paraf-structure and proves that a manifoldM˜endowed with such a structure isξ-Ricci flat and is foliated by minimal hypersurfaces normal toξ, which are of Otsuki's type. Further one considers onM˜a2(n−1)-dimensional involutive distributionP⊥and a recurrent vector fieldV˜. It is proved that the maximal integral manifoldM⊥ofP⊥hasVas the mean curvature vector (up to1/2(n−1)). If the complimentary orthogonal distributionPofP⊥is also involutive, then the whole manifoldM˜is foliate. Different other properties regarding the vector fieldV˜are discussed.

scholarly journals Hopf-type theorem for self-shrinkers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hilário Alencar ◽  
Gregório Silva Neto ◽  
Detang Zhou
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Type Theorem ◽  
Complex Function ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Dimensional Euclidean Space ◽  
Position Vector ◽  
Weighted Mean Curvature ◽  
Radial Weight

Abstract In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three-dimensional Euclidean space ℝ 3 {{\mathbb{R}}^{3}} is a round sphere, provided its mean curvature and the norm of the its position vector have an upper bound in terms of the norm of its traceless second fundamental form. The example constructed by Drugan justifies that the hypothesis on the second fundamental form is necessary. We can also prove the same kind of rigidity results for surfaces with parallel weighted mean curvature vector in ℝ n {{\mathbb{R}}^{n}} with radial weight. These results are applications of a new generalization of Cauchy’s Theorem in complex analysis which concludes that a complex function is identically zero or its zeroes are isolated if it satisfies some weak holomorphy.

FREE ENERGY AND THE STATIC POTENTIAL FOR OPEN SMOOTH STRINGS

1988 ◽  
Vol 03 (09) ◽  
pp. 2195-2206 ◽  
Author(s):  
K.S. VISWANATHAN ◽  
ZHOU XIAOAN
Keyword(s):  
Free Energy ◽  
Path Integral ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Static Potential ◽  
Negative Norm ◽  
Higher Derivative ◽  
The Mean

The methods of Polchinski, and Burgess and Morris are used and extended to evaluate Polyakov’s path integral for open, oriented smooth strings on a cylinder. The smooth string action possesses an (on-shell) invariance under normal variations in the direction of the mean curvature vector of the imbedded surface provided the surface is stationary. Fixing this gauge in the path integral allows one to eliminate all negative norm states arising from higher derivative terms. The free energy and the static potential of the smooth strings are computed. We find that the open smooth strings can be made tachyon-free and has the preferred coefficient −π/6 for the 1/R term in the static potential (for d=4) for large R.

scholarly journals Surfaces with Mean Curvature Vector Parallel in the Normal Bundle

1972 ◽  
Vol 47 ◽  
pp. 161-167 ◽  
Author(s):  
Bang-Yen Chen ◽  
Gerald D. Ludden
Keyword(s):  
Mean Curvature ◽  
Tangent Space ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Normal Space ◽  
Mean Curvature Vector ◽  
Bilinear Mapping ◽  
The Second Fundamental Form ◽  
Connected Surface ◽  
The Mean

Let M be a connected surface immersed in a Euclidean m-space Em. Let h be the second fundamental form of this immersion it is a certain symmetric bilinear mapping for X ∈ M, where Tx is the tangent space and the normal space of M at x. Let H be the mean curvature vector of M in Em. If there exists a real λ such that for all tangent vectors X, Y in Tx, then ilf is said to be pseudo-umbilical at x.

SURFACES IN Sn WITH PRESCRIBED GAUSS MAP AND MEAN CURVATURE VECTOR FIELD

2006 ◽  
Vol 17 (10) ◽  
pp. 1127-1143
Author(s):  
AYAKO TANAKA
Keyword(s):  
Vector Field ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Curvature Vector ◽  
Gauss Map ◽  
Necessary And Sufficient Conditions ◽  
Mean Curvature Vector ◽  
The Mean ◽  
Necessary And Sufficient

We give relations between the Gauss map and the mean curvature vector field of a surface in the Euclidean unit n-sphere Sn. These relations are necessary and sufficient conditions for the existence of a surface in Sn with prescribed Gauss map and mean curvature vector field. We show that such surfaces can be expressed explicitly by using given data.

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