共查询到20条相似文献,搜索用时 453 毫秒
1.
Using the method of moving frames, we prove that any irreducible Dupin hypersurface in S
5 with four distinct principal curvatures and constant Lie curvature is equivalent by Lie sphere transformation to an isoparametric hypersurface in S
5. 相似文献
2.
Yu-Chung Chang 《Monatshefte für Mathematik》2009,4(4):1-22
In this paper we consider a compact oriented hypersurface M
n
with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S
n+1. Denote by fij{\phi_{ij}} the trace free part of the second fundamental form of M
n
, and Φ be the square of the length of fij{\phi_{ij}} . We obtain two integral formulas by using Φ and the polynomial
PH,m(x)=x2+ \fracn(n-2m)?{nm(n-m)}H x -n(1+H2){P_{H,m}(x)=x^{2}+ \frac{n(n-2m)}{\sqrt{nm(n-m)}}H x -n(1+H^{2})} . Assume that B
H,m
is the square of the positive root of P
H,m
(x) = 0. We show that if M
n
is a compact oriented hypersurface immersed in the sphere S
n+1 with constant mean curvatures H having two distinct principal curvatures λ and μ then either F = BH,m{\Phi=B_{H,m}} or F = BH,n-m{\Phi=B_{H,n-m}} . In particular, M
n
is the hypersurface Sn-m(r)×Sm(?{1-r2}){S^{n-m}(r)\times S^{m}(\sqrt{1-r^{2}})} . 相似文献
3.
Using the method of moving frames, we prove that any locally irreducible Dupin hypersurface in S
n
with three distinct principal curvatures is equivalent by Lie sphere transformation to an isoparametric hypersurface in S
n
.
Oblatum VI-1995 & 28-IV-1997 相似文献
4.
Yu-Chung Chang 《Monatshefte für Mathematik》2009,158(1):1-22
In this paper we consider a compact oriented hypersurface M
n
with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S
n+1. Denote by the trace free part of the second fundamental form of M
n
, and Φ be the square of the length of . We obtain two integral formulas by using Φ and the polynomial . Assume that B
H,m
is the square of the positive root of P
H,m
(x) = 0. We show that if M
n
is a compact oriented hypersurface immersed in the sphere S
n+1 with constant mean curvatures H having two distinct principal curvatures λ and μ then either or . In particular, M
n
is the hypersurface .
相似文献
5.
A hypersurface x : M → S n+1 without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = ?ρ ?2∑ i (e i (H) + ∑ j (h ij ?Hδ ij )e j (log ρ))θ i vanishes and its Möbius shape operator $ {\Bbb {S}}A hypersurface x : M → S
n
+1 without umbilic point is called a M?bius isoparametric hypersurface if its M?bius form Φ = −ρ−2∑
i
(e
i
(H) + ∑
j
(h
ij
−Hδ
ij
)e
j
(log ρ))θ
i
vanishes and its M?bius shape operator ? = ρ−1(S−Hid) has constant eigenvalues. Here {e
i
} is a local orthonormal basis for I = dx·dx with dual basis {θ
i
}, II = ∑
ij
h
ij
θ
i
⊗θ
i
is the second fundamental form, and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S
n
+1 is a M?bius isoparametric hypersurface, but the converse is not true. In this paper we classify all M?bius isoparametric
hypersurfaces in S
n
+1 with two distinct principal curvatures up to M?bius transformations. By using a theorem of Thorbergsson [1] we also show
that the number of distinct principal curvatures of a compact M?bius isoparametric hypersurface embedded in S
n
+1 can take only the values 2, 3, 4, 6.
Received September 7, 2001, Accepted January 30, 2002 相似文献
6.
Huiling Le 《Probability Theory and Related Fields》1999,114(1):85-96
Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m≥ 3 and that, outside a fixed compact set, the sectional curvatures are bounded above by −c
1/{r
2 ln r} and below by −c
2
r
2, where c
1 and c
2 are two positive constants and r is the geodesic distance from a fixed point. We show that, when κ≥ 1 satisfies certain conditions, the angular part of a
κ-quasi-conformal Γ-martingale on M tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire
sphere at infinity. This improves both a result of Le for Brownian motion and also results concerning the non-existence of
κ-quasi-conformal harmonic maps from certain types of Riemannian manifolds into M.
Received: 19 September 1997 相似文献
7.
Let M be a closed Willmore hypersurface in the sphere S^n+1(1) (n ≥ 2) with the same mean curvature of the Willmore torus Wm,n-m, if SpecP(M) = Spec^P(Wm,n-m ) (p = 0, 1,2), then M is Wm,n-m. 相似文献
8.
The purpose of this paper is to make clear the so-called Nomizu problem, whether it is possible to find examples of space-like
isoparametric hypersurfaces in H
1
n+1 with more than two distinct principal curvatures. It is proved that a space-like isoparametric hypersurface in H
1
n+1 or S
1
n+1 can have at most two distinct principal curvatures. The authors present the classification and explicit analytic expressions
of such type of isoparametric hypersurfaces.
This paper was translated from J. Nanchang Univ. Nat. Sci. Ed., 2004, 28(2): 113–117 相似文献
9.
We consider a proper, umbilic-free immersion of an n-dimensional manifold M in the sphere S
n+1. We show that M is a Moebius isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Moebius
curvature. 相似文献
10.
Let x:M→ be an isometric immersion of a hypersurface M into an (n+1)-dimensional Riemannian manifold and let ρ
i
(i∈{1,...,n}) be the principal curvatures of M. We denote by E and P the distinguished vector field and the curvature vector field of M, respectively, in the sense of [8].?If M is structured by a P-parallel connection [7], then it is Einsteinian. In this case, all the curvature 2-forms are exact and other properties induced
by E and P are stated.?The principal curvatures ρ
i
are isoparametric functions and the set (ρ1,...,ρ
n
) defines an isoparametric system [10].?In the last section, we assume that, in addition, M is endowed with an almost symplectic structure. Then, the dual 1-form π=P
♭ of P is symplectic harmonic. If M is compact, then its 2nd Betti number b
2≥1.
Received: April 7, 1999; in final form: January 7, 2000?Published online: May 10, 2001 相似文献
11.
In this work we generalize the case of scalar curvature zero the results of Simmons (Ann. Math. 88 (1968), 62–105) for minimal cones in Rn+1. If Mn−1 is a compact hypersurface of the sphere Sn(1) we represent by C(M)ε the truncated cone based on M with center at the origin. It is easy to see that M has zero scalar curvature if and only if the cone base on M also has zero scalar curvature. Hounie and Leite (J. Differential Geom. 41 (1995), 247–258) recently gave the conditions for the ellipticity of the partial differential equation of the scalar curvature.
To show that, we have to assume n ⩾ 4 and the three-curvature of M to be different from zero. For such cones, we prove that, for n ≤slant 7 there is an ε for which the truncate cone C(M)ε is not stable. We also show that for n ⩾ 8 there exist compact, orientable hypersurfaces Mn−1 of the sphere with zero scalar curvature and S3 different from zero, for which all truncated cones based on M are stable.
Mathematics Subject Classifications (2000): 53C42, 53C40, 49F10, 57R70. 相似文献
12.
In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to
compact spacelike hypersurfaces which are immersed in de Sitter spaceS
1
n+1
(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant
scalar curvaturen(n−1)r is isometric to a sphere ifr<c.
Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and
Culture. 相似文献
13.
Majid Ali Choudhary 《Russian Mathematics (Iz VUZ)》2014,58(8):56-64
S. Deshmukh has obtained interesting results for first nonzero eigenvalue of a minimal hypersurface in the unit sphere. In the present article, we generalize these results to pseudoumbilical hypersurface and prove: What conditions are satisfied by the first nonzero eigenvalue λ 1 of the Laplacian operator on a compact immersed pseudo-umbilical hypersurface M in the unit sphere S n+1. We also show that a compact immersed pseudo-umbilical hypersurface of the unit sphere S n+1 with λ 1 = n is either isometric to the sphere S n or for this hypersurface an inequaluity is fulfilled in which sectional curvatures of the hypersuface M participate. 相似文献
14.
In the Euclidean Space
\mathbb Rn+1{\mathbb {R}^{n+1}} with a density
ee\frac12 n m2 |x|2, (e = ±1){e^{\varepsilon \frac12 n \mu^2 |x|^2},} {(\varepsilon =\pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account
of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ \varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere)
depending on the relation between the bound of its principal curvatures and μ. 相似文献
15.
Stefan Immervoll 《Archiv der Mathematik》2006,87(5):478-480
We generalize a result of Kramer, see [7, 10.7 and 10.10], on generalized quadrangles associated with isoparametric hypersurfaces
of Clifford type to Tits buildings of type C2 derived from arbitrary isoparametric hypersurfaces with four distinct principal curvatures in spheres: two distinct points
p and q of a generalized quadrangle associated with an isoparametric hypersurface in the unit sphere of a Euclidean vector space
can be joined by a line K if and only if (p − q)/||p − q|| is a line. This line is orthogonal to K. Dually, two distinct lines L and K intersect if and only if (L − K)/||L − K|| is point.
Received: 14 October 2005 相似文献
16.
To a given immersion
i:Mn? \mathbb Sn+1{i:M^n\to \mathbb S^{n+1}} with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|2. We prove the existence of a constant C
n
(R) depending on R and n so that R ≥ 1 and sup |A|2 = C
n
(R) imply that the hypersurface is a H(r)-torus
\mathbb S1(?{1-r2})×\mathbb Sn-1 (r){\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}. For R > (n − 2)/n we use rotation hypersurfaces to show that for each value C > C
n
(R) there is a complete hypersurface in
\mathbb Sn+1{\mathbb S^{n+1}} with constant scalar curvature R and sup |A|2 = C, answering questions raised by Q. M. Cheng. 相似文献
17.
We consider a proper, umbilic-free immersion of an n-dimensional manifold M in the sphere S n+1. We show that M is a Moebius isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Moebius curvature. 相似文献
18.
Spacelike hypersurfaces with constant scalar curvature 总被引:1,自引:0,他引:1
In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to
compact spacelike hypersurfaces which are immersed in de Sitter space S
n
+1
1(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant
scalar curvature n(n-1)r is isometric to a sphere if r << c.
Received: 18 December 1996 / Revised version: 26 November 1997 相似文献
19.
A hypersurface M
n
immersed in a space form is r-minimal if its (r + 1)
th
-curvature (the (r + 1)
th
elementary symmetric function of its principal curvatures) vanishes identically. Let W be the set of points which are omitted by the totally geodesic hypersurfaces tangent to M. We will prove that if an orientable hypersurface M
n
is r-minimal and its r
th
-curvature is nonzero everywhere, and the set W is nonempty and open, then M
n
has relative nullity n − r. Also we will prove that if an orientable hypersurface M
n
is r-minimal and its r
th
-curvature is nonzero everywhere, and the ambient space is euclidean or hyperbolic and W is nonempty, then M
n
is r-stable. 相似文献
20.
We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S n+1 satisfying Sf 4 f_3~2 ≤ 1/n S~3 , where S is the squared norm of the second fundamental form of M, and f_k =sum λ_i~k from i and λ_i (1 ≤ i ≤ n) are the principal curvatures of M. We prove that there exists a positive constant δ(n)(≥ n/2) depending only on n such that if n ≤ S ≤ n + δ(n), then S ≡ n, i.e., M is one of the Clifford torus S~k ((k/n)~1/2 ) ×S~... 相似文献