R. Kusner and N. Schmitt, The spinor representation of surfaces in space, Preprint, http://www.arxiv.org/abs/dg-ga/9610005, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1 (CMC 1 surfaces) This lead Bryant [Br] to derive a representation for CMC 1 surfaces in terms of holomorphic data. The holomorphic data used in the Weierstrass representation for minimal surfaces consists alternatively of a function and a one form, or of two spinors with the same spin structure [Bo, KS]. These functions, forms, and spinors are de ned on the same Riemann surface as the conformal minimal immersion which they represent. Bryant s representation for CMC 1 surfaces also involves two spinors with the same spin structure. Other researchers prefer an equivalent version involving a ....

R. Kusner, N. Schmitt, The spinor representation of surfaces in space, arXiv:dg-ga/9610005, (1996) 28

....spin structures to the set of spin structures. For M = T 2 , the trivial spin structure is characterized by the fact that for any non contractible embedding S 1 T 2 the tangent vector field c : S 1 TT 2 lifts to T 2 . The line bundle spin structure definition is used by [KS97] for example. The Arf invariant [KS97] can also be used to distinguish the trivial spin structure from the non trivial ones. The Arf invariant is equal to 1 for the trivial spin structure, and equal to 1 for all others. 5 Comparing spectra of conformal manifolds In this section we will compare ....

....structures. For M = T 2 , the trivial spin structure is characterized by the fact that for any non contractible embedding S 1 T 2 the tangent vector field c : S 1 TT 2 lifts to T 2 . The line bundle spin structure definition is used by [KS97] for example. The Arf invariant [KS97] can also be used to distinguish the trivial spin structure from the non trivial ones. The Arf invariant is equal to 1 for the trivial spin structure, and equal to 1 for all others. 5 Comparing spectra of conformal manifolds In this section we will compare Dirac and Laplace eigenvalues on 2 tori. ....

R. Kusner and N. Schmitt. The spinor representation of surfaces in space. Preprint, 1997.

....isomorphism so(3) su(2) and to rewrite the equations (3, 4) for the moving frame in terms of 2 by 2 matrices. This quaternionic description turns out to be useful for analytic studies of general curves and surfaces in 3 and 4 spaces as well as for investigation of special classes of surfaces [Bob1, KS2, DPW, Bob2, KPP, PP]. Let us denote the algebra of quaternions by H , the multiplicative quaternion group by H = H n f0g, and their standard basis by f1; i; j; kg, where ij = k; jk = i; ki = j: 9) This basis can be represented by the Pauli matrices as follows: 1 = 0 1 1 0 = i i; 2 = 0 i i 0 ....

....out that at this point the whole construction can be reversed. Namely, starting with a solution to the Dirac equation one can derive a Weierstrass type representation (see (21) below) for conformally parametrized surfaces. This idea was recently developed by Konopelchenko [Kon] and further in [Tai, PP, KS2], although in other forms the Weierstrass representation of surfaces was known already to Eisenhart [Eis] and Kenmotsu [Ken] Theorem 3 Let D C be a simply connected domain and (s 1 ; s 2 ) T : D C 2 be a solution to the Dirac equation with the potential p 2 C 1 (D) 0 z z ....

[Article contains additional citation context not shown here]

Kusner, R., Schmidt, N.: The spinor representation of surfaces in space, GANG Preprint IV.18 (1996)

No context found.

R. Kusner and N. Schmitt, The spinor representation of surfaces in space, Preprint, http://www.arxiv.org/abs/dg-ga/9610005, 1996.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC