Integral Geometry in Homogeneous Spaces

Abstract

f H°[A r\g(B)]dHk+l+mg = yHk(A)Hl(B).If instead of compact G, one has X = R" and G = R" x 0", then f f H°(An[z+g(B)])dHk + lzd<J>"g = yH\A)H\B), Jo"Jr" where <¡?n is a Haar measure on 0".y is a constant depending only upon A0 and B0.y>0 if and only if for some geG there exists aeA0(~,g(B0) for which Ta(A(f) C\ Ta[g(B0)]=0.This theorem is a special case of 11.1 and is closely related to a result of Freilich [14].In proving this theorem, Federer's coarea formula is generalized to give nontrivial results for maps f:X-*Y such that rankf#(x) < dim Y for x e X.If X has constant curvature, a and y are independent of A0,B0.Now suppose k+l¡zn.In [12] Fédérer developed a theory which can be used to define in a natural manner the intersections of a normal k current with either the isometric images of a normal I current or the elements of S. One may therefore seek integralgeometric formulas for normal currents.In order to define these intersections, a "lifting" map Lm is defined and studied for fibre bundles Ha having coherently oriented fibres.La is a chainmonomorphism of the complex of normal currents in the base space into the complex of normal currents in the bundle space, continuous on N bounded sets.Suppose X is oriented and has constant curvature, S is a normal k current in X and T is a normal 1 current in X.Then S ng# (T) is a normal k + I -n current for almost all geG and there exists a constant ô>0, depending only upon k, I and T, such that for each bounded Baire k form ep on X and each bounded Baire I form ip on X, js^g0(T) (*[*eb A*g~1* W])dVg CÔS(ep)T(ib) if k > n/2 or / > n/2, \b[S(ep)T(\\>) + S(*e/>) T(*i/0] if k = / = n/2.Now assume that the elements of S are oriented and for E e S G n {g :g#iE) = É} acts transitively on F. Then S C\E is a normal k+ I -n current for almost all Eeef, and there exists a constant e > 0, depending only upon k, I and <D, such that S O E[*i*ep A *\!/E)]d<&E = eSiep) for each bounded Baire k form ep on X. [\//E is the covariant dual of the positively oriented unit I vector field on E.] These statements are also true for quasi-normal currents.I

Locations

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