Geometric & Functional Analysis GAFA

, Volume 11, Issue 3, pp 589–608

Polar factorization of maps on Riemannian manifolds

  • R.J. McCann

DOI: 10.1007/PL00001679

Cite this article as:
McCann, R. GAFA, Geom. funct. anal. (2001) 11: 589. doi:10.1007/PL00001679


Let (M,g) be a connected compact manifold, C3 smooth and without boundary, equipped with a Riemannian distance d(x,y). If \( s : M \to M \) is merely Borel and never maps positive volume into zero volume, we show \( s = t \circ u \) factors uniquely a.e. into the composition of a map \( t(x) = {\rm exp}_x[-\nabla\psi(x)] \) and a volume-preserving map \( u : M \to M \), where \( \psi : M \to {\bold R} \) satisfies the additional property that \( (\psi^c)^c = \psi \) with \( \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} \) and c(x,y) = d2(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.¶The results are obtained by solving a Riemannian version of the Monge—Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution \( f \ge 0 \) of mass in L1(M) onto another. Parallel results for other strictly convex cost functions \( c(x,y) \ge 0 \) of the Riemannian distance on non-compact manifolds are briefly discussed.

Copyright information

© Birkhäuser Verlag, Basel 2001

Authors and Affiliations

  • R.J. McCann
    • 1
  1. 1.Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3, e-mail: mccann@math.toronto.eduCA

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