Realization of Chiral Symmetry in a Curved Isospin Space

The nonlinear realizations of the chiral group SU(2)⊗SU(2) are studied from a geometric point of view. The three-dimensional nonlinear realization, associated with the pion field, is considered as a group of coordinate transformations in a three-dimensional isospin space of constant curvature, leaving invariant the line element. Spinor realizations in general coordinates are constructed by combined coordinate-spin-space transformations in analogy to Pauli's method for spinors in general relativity. The description of vector mesons and possible chiral-invariant Lagrangians, which yield the various nonlinear models in specific frames of general coordinates, are discussed.