Symmetry in Nonlinear Mathematical Physics - 2009

Olga Kunakovskaya (Voronezh State University, Russia)
Alfonso Vignoli (Universita degli Studi di Roma "Tor Vergata", Italy)
Antonio Vitolo (Universita degli Studi di Salerno, Italy)

Equivariant vector fields and their indices

We introduce Γ-equivariant local and global (total) indices of connected components of the zero set of a smooth Γ-equivariant (co)tangent vector field on a compact n-dimensional smooth manifold (possibly with boundary) with a given smooth action of a compact Lie group Γ. These indices belong to Γ-equivariant homotopy groups of spheres. The basis of our construction is the Ize–Vignoli degree theory of Γ-equivariant maps. Two existence theorems of solutions for nonlinear problems of Equivariant Analysis are obtained.