Realization of Chiral Symmetry in a Curved Isospin Space

The nonlinear realizations of the chiral group $\text{SU(2)⊗SU(2)}$ 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.