scholarly journals Expanding 3d $$ \mathcal{N} $$ = 2 theories around the round sphere

2020 ◽  
Vol 2020 (2) ◽  
Author(s):  
Dongmin Gang ◽  
Masahito Yamazaki
Keyword(s):  
Round Sphere

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 CPn, OR, ENTANGLEMENT ILLUSTRATED

2002 ◽  
Vol 17 (31) ◽  
pp. 4675-4695 ◽  
Author(s):  
INGEMAR BENGTSSON ◽  
JOHAN BRÄNNLUND ◽  
KAROL ŻYCZKOWSKI
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Round Sphere ◽  
Geometrical Properties ◽  
Flat Tori ◽  
Complex Projective

We show that many topological and geometrical properties of complex projective space can be understood just by looking at a suitably constructed picture. The idea is to view CPn as a set of flat tori parametrized by the positive octant of a round sphere. We pay particular attention to submanifolds of constant entanglement in CP3 and give a few new results concerning them.

scholarly journals Localization on round sphere revisited

2013 ◽  
Vol 2013 (11) ◽  
Author(s):  
Akinori Tanaka
Keyword(s):  
Round Sphere

THE ADAPTED COMPLEXIFICATION OF AN ELLIPSOID

2007 ◽  
Vol 18 (01) ◽  
pp. 43-68
Author(s):  
RAUL M. AGUILAR
Keyword(s):  
Geodesic Flow ◽  
Riemannian Metric ◽  
Classical Theorem ◽  
Round Sphere

We show that an n-dimensional real ellipsoid in ℝn+1 with the induced Riemannian metric does not admit an unbounded adapted complexification in the sense of Lempert/Szőke and Guillemin/Stenzel, unless it is a round sphere. In other words, an ellipsoid whose (maximal) Grauert tube has infinite radius must be a round sphere. For the proof we take advantage of the integrability of the geodesic flow and use a classical theorem on umbilic geodesics. We carry out an extension of this result to Liouville metrics elsewhere.

Trapped submanifolds contained into a null hypersurface of de Sitter spacetime

2018 ◽  
Vol 20 (08) ◽  
pp. 1750059 ◽  
Author(s):  
Luis J. Alías ◽  
Verónica L. Cánovas ◽  
Marco Rigoli
Keyword(s):  
Steady State ◽  
State Space ◽  
Light Cone ◽  
De Sitter Spacetime ◽  
De Sitter ◽  
Round Sphere ◽  
Codimension Two ◽  
The Past ◽  
Sitter Spacetime ◽  
Spacelike Submanifolds

We study codimension two trapped submanifolds contained into one of the two following null hypersurfaces of de Sitter spacetime: (i) the future component of the light cone, and (ii) the past infinite of the steady state space. For codimension two compact spacelike submanifolds in the light cone we show that they are conformally diffeomorphic to the round sphere. This fact enables us to deduce that the problem of characterizing compact marginally trapped submanifolds into the light cone is equivalent to solving the Yamabe problem on the round sphere, allowing us to obtain our main classification result for such submanifolds. We also fully describe the codimension two compact marginally trapped submanifolds contained into the past infinite of the steady state space and characterize those having parallel mean curvature field. Finally, we consider the more general case of codimension two complete, non-compact, weakly trapped spacelike submanifolds contained into the light cone.

scholarly journals Round spheres are Hausdorff stable under small perturbation of entropy

2020 ◽  
Vol 2020 (758) ◽  
pp. 261-280
Author(s):  
Shengwen Wang
Keyword(s):  
Small Perturbation ◽  
Hausdorff Distance ◽  
Higher Dimensions ◽  
Round Sphere ◽  
Closed Hypersurface

AbstractWe show that if the entropy of any closed hypersurface is close to that of a round hyper-sphere, then it is close to a round sphere in Hausdorff distance. This generalizes the result of [3] to higher dimensions.

scholarly journals THE ENERGY-MOMENTUM TENSOR ON LOW DIMENSIONAL Spinc MANIFOLDS

2012 ◽  
Vol 23 (09) ◽  
pp. 1250090 ◽  
Author(s):  
GEORGES HABIB ◽  
ROGER NAKAD
Keyword(s):  
Energy Momentum Tensor ◽  
Type Inequality ◽  
Momentum Tensor ◽  
Compact Surface ◽  
Round Sphere ◽  
Spinor Equation ◽  
Immersed Surfaces ◽  
Low Dimensional ◽  
Limiting Case

On a compact surface endowed with any Spinc structure, we give a formula involving the Energy-Momentum tensor in terms of geometric quantities. A new proof of a Bär-type inequality for the eigenvalues of the Dirac operator is given. The round sphere 𝕊2 with its canonical Spinc structure satisfies the limiting case. Finally, we give a spinorial characterization of immersed surfaces in 𝕊2 × ℝ by solutions of the generalized Killing spinor equation associated with the induced Spinc structure on 𝕊2 × ℝ.

scholarly journals Does negative type characterize the round sphere?

2007 ◽  
Vol 135 (11) ◽  
pp. 3695-3703 ◽  
Author(s):  
Simon Lyngby Kokkendorff
Keyword(s):  
Round Sphere ◽  
Negative Type

scholarly journals New Characterizations of the Clifford Torus and the Great Sphere

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1076 ◽  
Author(s):  
Sun Mi Jung ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Euclidean Space ◽  
Finite Type ◽  
Three Dimensional ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Dimensional Sphere ◽  
Round Sphere ◽  
Clifford Torus ◽  
Great Sphere ◽  
Set Up

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we study ruled surfaces in a three-dimensional sphere with finite-type and pointwise 1-type spherical Gauss map. Concerning integrability and geometry, we set up new characterizations of the Clifford torus and the great sphere of 3-sphere and construct new examples of spherical ruled surfaces in a three-dimensional sphere.

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