About Systems of Differential Equations Related to Geodesic Double Differential Forms Mean Values and Harmonic Spaces Part I

In spaces of constant curvature P. Gunther has introduced geodesic double differential forms and mean value operators for differential forms. As solutions of differential equations for forms (for instance, Weyl-De Rham equations) the form equations so as the mean values suffice certain ordinary systems of differential equations. We generalize such systems and study properties which are determined by differential geometric structures (parallel translation of double differential forms with respect to geodesic lines, the construction of closed, coclosed and harmonic components of differential forms and double forms and the telescopage theorem of McKean and Singer). As special cases we get the systems known for geodesic double differential forms or mean value operators in real and complex hyperbolic spaces by the application of structural information without any special information about these spaces.