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Hayashi J. Math. Phys. 46, 022101 (2005); quant-ph/0406038. 10. M. Nakahara, Y. Kondo, K. Hata and S. Tanimura, Phys. Rev. A70, 052319 (2004); quant-ph/0405050. 11. M. Nakahara, J. J. Vartiainen, Y. Kondo, S. Tanimura and K. Hata; quantph/0411153. 12. Y. Kondo, M. Nakahara, K. Hata and S. Tanimura; quant-ph/0503067. II Topological Crystals 35 TOPOLOGICAL CRYSTALS OF NbSe ; SATOSHI TANDA1, TAKU TSUNETA2, TAKESHI TOSHIMA1, TORU MATSUURA1, AND MASAKATSU TSUBOTA1 Department of Applied Low Temperature Physics, Hokkaido University, 060-8628, Japan Laboratory, Helsinki Otakaari 3A, Espoo, University Finland Sapporo, of Hokkaido Technology, We report the discovery of a Mobius crystal of NbSe3, conventionally grown as ribbons and whiskers.

Tanda, T. Tsuneta, Y. Okajima, K. Inagaki, K. Yamaya and N. Hatakenaka, Nature 417, 397 (2002). 16 TOPOLOGY IN PHYSICS* R. edu The phenomenon of quantum number fractionalization is explained. The relevance of non-trivial phonon field topology is emphasized. 1. Introduction Discussions of the spatial forms of physical materials use in a natural way geometrical and topological concepts. It is to be expected that arrangements of matter should form patterns that are described by pre-existing mathematical structures drawn from geometry and topology.

The canonical connection form on S,jv,fc(C) is defined as a one-form A = VUV, (12) which takes its value in the Lie algebra u(k). The holonomy associated with this connection is called the Berry phase in case of k = 1 and the Wilczek-Zee holonomy in general. We define Riemannian metrices, ||<2V||2 = tr(dVUV) for the Stiefel manifold and ||dP|| 2 = tr (dPdP) for the Grassmann manifold. For any curve P(t) in Gjv,fc(C), there is a curve V(t) in SN,k(C) such that n(V(t)) = P{t). f-0, it is called a horizontal lift of the curve P(t).