Spinorial characterizations of surfaces into three-dimensional homogeneous manifolds

Roth, Julien

doi:10.1016/j.geomphys.2010.03.007

Cited by 22 publications

(57 citation statements)

References 14 publications

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“…As in [5] (and after in [2,3,14,15] in codimension one, and in [6] and [7] in codimension two) the proof of this proposition relies on the fact that such a spinor field is necessarily a solution of (33), with this bilinear map B: …”

Section: Satisfies the Gauss Codazzi And Ricci Equations And Is Suchmentioning

confidence: 98%

Spinor representation of Lorentzian surfaces in R2,2

Bayard

Patty

2015

Journal of Geometry and Physics

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a b s t r a c tWe prove that an isometric immersion of a simply connected Lorentzian surface in R 2,2 is equivalent to a normalised spinor field solution of a Dirac equation on the surface. Using the quaternions and the Lorentz numbers, we also obtain an explicit representation formula of the immersion in terms of the spinor field. We then apply the representation formula in R 2,2 to give a new spinor representation formula for Lorentzian surfaces in 3-dimensional Minkowski space. Finally, we apply the representation formula to the local description of the flat Lorentzian surfaces with flat normal bundle and regular Gauss map in R 2,2 , and show that these surfaces locally depend on four real functions of one real variable, or on one holomorphic function together with two real functions of one real variable, depending on the sign of a natural invariant.

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Section: Satisfies the Gauss Codazzi And Ricci Equations And Is Suchmentioning

confidence: 98%

Spinor representation of Lorentzian surfaces in R2,2

Bayard

Patty

2015

Journal of Geometry and Physics

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“…Finally, we give a spinorial version of this theorem as in the Riemannian case [6,11,14] and generalizing the results of [8,9] to three-dimensional Lorentzian products.…”

Section: Then There Exists An Isometric Immersionmentioning

confidence: 94%

“…Using computations of [9,14], we can prove that (18) is equivalent to (21) and (22). Hence, as in [14], Theorem 3 can be rewritten with two orthogonal spinors solutions of the Dirac equation (21) and satisfying condition (22).…”

Section: Isometric Immersions Into Lorentzian Products 1285mentioning

confidence: 99%

“…These results are the geometrically invariant version of previous work on the spinorial Weierstrass representation by Abresch, Sullivan, Kusner, Schmidt, and many others (see [7]). This representation was expressed by Friedrich [6] for surfaces in R 3 and then extended to other three-dimensional Riemannian manifolds (see [11,14]). In a recent work, we have treated this question for surfaces of arbitrary signature in pseudo-Riemannian three-dimensional space forms [9].…”

Section: Spinorial Characterization Of Surfaces Intomentioning

confidence: 99%

“…Hence, as in [14], Theorem 3 can be rewritten with two orthogonal spinors solutions of the Dirac equation (21) and satisfying condition (22).…”

Section: Isometric Immersions Into Lorentzian Products 1285mentioning

confidence: 99%

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Isometric Immersions Into Lorentzian Products

Roth¹

2011

Int. J. Geom. Methods Mod. Phys.

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We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed into one of the Lorentzian products S n ×R 1 or H n ×R 1 . This condition is expressed in terms of its first and second fundamental forms, the tangent and normal projections of the vertical vector field. As applications, we give an equivalent condition in a spinorial way and we deduce the existence of a one-parameter family of isometric maximal deformation of a given maximal surface obtained by rotating the shape operator. Isometric Immersions into Lorentzian Products 1271Moreover, this immersion is unique up to a global isometry of M n (κ) × R which preserves the orientation of R. Int. J. Geom. Methods Mod. Phys. 2011.08:1269-1290. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 07/07/15. For personal use only. 1272 J. Roth The compatibility equationsLet (M n , g) be an oriented Riemannian hypersurface of M n (κ) × R 1 with normal unit vector field ν. We denote by ∂ t the vertical vector of M n (κ) × R 1 , that is, the unit vector field giving the orientation of R 1 in M n (κ) × R 1 . The projection of ∂ t on T M is denoted by T and its normal part is f . Therefore, we have ∂ t = T + f ν, with f = − ∂ t , ν since ν, ν = −1. We can compute the curvature tensor of M n (κ)× R 1 for vector fields tangent to M . We have the following proposition. Proposition 2.1. For any vector fields X, Y, Z, W ∈ Γ(T M), we haveProof. Let X, Y, Z, W ∈ Γ(T M). We can write these vector fields as follows:with X, Y , Z and W tangent to M n (κ) and x, y, z and w some real-valued functions.Since ∂ t is parallel, we haveMoreover, since the vector fields X, Y, Z and W are tangent to M and ∂ t , ∂ t = −1, we have x = − X, T , y = − Y, T , z = − Z, T and w = − W, T . So, a straightforward computation yields the first identity. For the second identity, we denote, ν = ν + n∂ t with ν tangent to M n (κ). Clearly, we see that n = − ν, ∂ t = f . So, we havewhich gives easily the wanted identity. Int. J. Geom. Methods Mod. Phys. 2011.08:1269-1290. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 07/07/15. For personal use only.

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Spinorial Representation of Submanifolds in $${\varvec{SL}}_{\varvec{n}}{\varvec{(\mathbb {C})/SU(n)}}$$

Bayard

2019

Adv. Appl. Clifford Algebras

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Spinorial characterizations of surfaces into three-dimensional homogeneous manifolds

Cited by 22 publications

References 14 publications

Spinor representation of Lorentzian surfaces in R2,2

Spinor representation of Lorentzian surfaces in R2,2

Isometric Immersions Into Lorentzian Products

Spinorial Representation of Submanifolds in $${\varvec{SL}}_{\varvec{n}}{\varvec{(\mathbb {C})/SU(n)}}$$

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