We study harmonic maps from Riemann surfaces M to the loop spacesΩG of compact Lie groups G,using the twistor approach.We conjecture that harmonic maps of the Riemann sphere CP^1 intoΩG are related to Yang-Mills G-f...We study harmonic maps from Riemann surfaces M to the loop spacesΩG of compact Lie groups G,using the twistor approach.We conjecture that harmonic maps of the Riemann sphere CP^1 intoΩG are related to Yang-Mills G-fields on R^4.展开更多
Over an oriented even dimensional Riemannian manifold(M 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description ...Over an oriented even dimensional Riemannian manifold(M 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description of the twistor bundle Гm = SO(2m)/U(m)? J +(@#@ M,ds2 ) →M. The integrability on an almost complex structureJ compatible with the metric and the orientation, is shown to be equivalent to the fact that the corresponding cross section of the twistor bundle is holomorphic with respect toJ and the canonical almost complex structureJ 1 onJ +(M,ds2 ), by using moving frame theory. Moreover, for various metrics and a fixed orientation onM, a canonical bundle isomorphism is established. As a consequence, we generalize a celebrated theorem of LeBrun.展开更多
基金This work was partly supported by the RFBR(Grant Nos.04-01-00236,06-02-04012)by the program of Support of Scientific Schools(Grant No.1542.2003.1)by the Scientific Program of RAS"Nonlinear Dynamics"
文摘We study harmonic maps from Riemann surfaces M to the loop spacesΩG of compact Lie groups G,using the twistor approach.We conjecture that harmonic maps of the Riemann sphere CP^1 intoΩG are related to Yang-Mills G-fields on R^4.
基金This work was supported partially by the National Natural Science Foundation of China(Grant No.10131020)Outstanding Youth Foundation of China No.19925103 and No.10229101the“973”.
文摘Over an oriented even dimensional Riemannian manifold(M 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description of the twistor bundle Гm = SO(2m)/U(m)? J +(@#@ M,ds2 ) →M. The integrability on an almost complex structureJ compatible with the metric and the orientation, is shown to be equivalent to the fact that the corresponding cross section of the twistor bundle is holomorphic with respect toJ and the canonical almost complex structureJ 1 onJ +(M,ds2 ), by using moving frame theory. Moreover, for various metrics and a fixed orientation onM, a canonical bundle isomorphism is established. As a consequence, we generalize a celebrated theorem of LeBrun.
