scholarly journals Darboux transformations for the twisted so(p, q) system and local isometric immersion of space forms

1998 ◽  
Vol 14 (5) ◽  
pp. 1353-1370 ◽  
Author(s):  
Zixiang Zhou
Keyword(s):  
Isometric Immersion ◽  
Darboux Transformations ◽  
Space Forms ◽  
Local Isometric Immersion ◽  
Q System

scholarly journals TOPOLOGICAL AND GEOMETRICAL PROPRIETIES OF BRANE-WORLDS

2011 ◽  
Vol 08 (05) ◽  
pp. 945-951
Author(s):  
EDMUNDO M. MONTE
Keyword(s):  
Symmetric Space ◽  
Isometric Immersion ◽  
Space Time ◽  
Brane World ◽  
Point Of View ◽  
Spherically Symmetric ◽  
Euclidean Spaces ◽  
Brane Worlds ◽  
Local Isometric Immersion

Through the characterization of a spherically symmetric space-time as a local brane-world immersed into six-dimensional pseudo-Euclidean spaces, with different signatures of the bulk, we investigate the existence of a topological difference in the immersed brane-world. In particular the Schwarzschild's brane-world and its Kruskal (or Frønsdal) brane-world extension are examined from point of view of the immersion formalism. We prove that there is a change of signature of the bulk when we consider a local isometric immersion and different topologies of a brane-world in that bulk.

scholarly journals GROWTH ESTIMATES FOR WARPING FUNCTIONS AND THEIR GEOMETRIC APPLICATIONS

2009 ◽  
Vol 51 (3) ◽  
pp. 579-592 ◽  
Author(s):  
BANG-YEN CHEN ◽  
SHIHSHU WALTER WEI
Keyword(s):  
Riemannian Manifold ◽  
Isometric Immersion ◽  
Warped Product ◽  
Minimal Immersion ◽  
Warping Function ◽  
Uniformization Theorem ◽  
Space Forms ◽  
Sharp Estimates ◽  
Geometric Curvature ◽  
Sectional Curvatures

AbstractBy applying Wei, Li and Wu's notion (given in ‘Generalizations of the uniformization theorem and Bochner's method in p-harmonic geometry’, Comm. Math. Anal. Conf., vol. 01, 2008, pp. 46–68) and method (given in ‘Sharp estimates on -harmonic functions with applications in biharmonic maps, preprint) and by modifying the proof of a general inequality given by Chen in ‘On isometric minimal immersion from warped products into space forms’ (Proc. Edinb. Math. Soc., vol. 45, 2002, pp. 579–587), we establish some simple relations between geometric estimates (the mean curvature of an isometric immersion of a warped product and sectional curvatures of an ambient m-manifold $\tilde M^m_c$ bounded from above by a non-positive number c) and analytic estimates (the growth of the warping function). We find a dichotomy between constancy and ‘infinity’ of the warping functions on complete non-compact Riemannian manifolds for an appropriate isometric immersion. Several applications of our growth estimates are also presented. In particular, we prove that if f is an Lq function on a complete non-compact Riemannian manifold N1 for some q > 1, then for any Riemannian manifold N2 the warped product N1 ×fN2 does not admit a minimal immersion into any non-positively curved Riemannian manifold. We also show that both the geometric curvature estimates and the analytic function growth estimates in this paper are sharp.

Explicit construction of Lagrangian isometric immersion of a real-space-form Mn(c) into a complex-space-form M˜n(4c)

2002 ◽  
Vol 132 (3) ◽  
pp. 481-508 ◽  
Author(s):  
YUN MYUNG OH
Keyword(s):  
Sectional Curvature ◽  
Isometric Immersion ◽  
Complex Space ◽  
Space Form ◽  
Real Space ◽  
Constant Sectional Curvature ◽  
Complex Space Form ◽  
Product Decomposition ◽  
Twisted Product ◽  
Space Forms

In [4], it is proved that there exists a ‘unique’ adapted Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c) of constant sectional curvature 4c associated with each twisted product decomposition of a real-space-form if its twistor form is twisted closed. Conversely, if L: Mn(c) → M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the immersion L is determined by the corresponding adapted Lagrangian isometric immersion of the twisted product decomposition. It is natural to ask the explicit expressions of adapted Lagrangian isometric immersions of twisted product decompositions of real-space-forms Mn(c) into complex-space-forms M˜n(4c) for each case: c = 0, c > 0 and c < 0.

scholarly journals CONSTANT SCALAR CURVATURE HYPERSURFACES WITH SECOND-ORDER UMBILICITY

2009 ◽  
Vol 51 (2) ◽  
pp. 219-241
Author(s):  
ANTONIO GERVASIO COLARES ◽  
FERNANDO ENRIQUE ECHAIZ-ESPINOZA
Keyword(s):  
Isometric Immersion ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Einstein Manifolds ◽  
Isometric Immersions ◽  
Principal Curvatures ◽  
Space Forms ◽  
Newton Polynomial ◽  
Product Of Spheres ◽  
The Mean

AbstractWe extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying that an isometric immersion is k-umbilical when APk−1(A) is a multiple of the identity, where Pk(A) is the kth Newton polynomial in the second fundamental form A with P0(A) being the identity. Thus, for k=1, one-umbilical coincides with umbilical. We determine the principal curvatures of the two-umbilical isometric immersions in terms of the mean curvatures. We give a description of the two-umbilical isometric immersions in space forms which includes the product of spheres $S^{k}(\frac{1}{\sqrt{2}})\times S^{k}(\frac{1}{\sqrt{2}})$ embedded in the Euclidean sphere S2k+1 of radius 1. We also introduce an operator φk which measures how an isometric immersion fails to be k-umbilical, giving in particular that φ1 ≡ 0 if and only if the immersion is totally umbilical. We characterize the two-umbilical hypersurfaces of a space form as images of isometric immersions of Einstein manifolds.

scholarly journals Local isometric immersions of pseudo-spherical surfaces and kth order evolution equations

2019 ◽  
Vol 21 (04) ◽  
pp. 1850025
Author(s):  
Nabil Kahouadji ◽  
Niky Kamran ◽  
Keti Tenenblat
Keyword(s):  
Isometric Immersion ◽  
Evolution Equations ◽  
Second Fundamental Form ◽  
Second Order ◽  
Isometric Immersions ◽  
Spherical Surfaces ◽  
Pseudospherical Surfaces ◽  
The Mean ◽  
Local Isometric Immersion ◽  
Straight Lines

We consider the class of evolution equations of the form [Formula: see text], [Formula: see text], that describe pseudo-spherical surfaces. These were classified by Chern and Tenenblat in [Pseudospherical surfaces and evolution equations, Stud. Appl. Math 74 (1986) 55–83.]. This class of equations is characterized by the property that to each solution of such an equation, there corresponds a 2-dimensional Riemannian metric of constant curvature [Formula: see text]. Motivated by the special properties of the sine-Gordon equation, we investigate the following problem: given such a metric, is there a local isometric immersion in [Formula: see text] such that the coefficients of the second fundamental form of the immersed surface depend on a jet of finite order of [Formula: see text]? We extend our earlier results for second-order evolution equations [N. Kahouadji, N. Kamran and K. Tenenblat, Local isometric immersions of pseudo-spherical surfaces and evolution equations, Fields Inst. Commun. 75 (2015) 369–381; N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, Comm. Anal. Geom. 24(3) (2016) 605–643.] to [Formula: see text]th order equations by proving that there is only one type of equation that admit such an isometric immersion. More precisely, we prove under the condition of finite jet dependency that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions [Formula: see text] are universal, i.e. they are independent of [Formula: see text]. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion.

Ideal Lagrangian immersions in complex space forms

2000 ◽  
Vol 128 (3) ◽  
pp. 511-533 ◽  
Author(s):  
BANG-YEN CHEN
Keyword(s):  
Riemannian Manifold ◽  
Isometric Immersion ◽  
Complex Space ◽  
Space Form ◽  
Lagrangian Submanifolds ◽  
Ambient Space ◽  
Complex Space Form ◽  
Space Forms ◽  
Complex Space Forms

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. In this paper we study Lagrangian immersions in complex space forms which are ideal. We prove that all Lagrangian ideal immersions in a complex space form are minimal. We also determine ideal Lagrangian submanifolds in complex space forms.

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