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CATKspaces and Alexandrov spaces
Totally geodesic maps
2 other sections not shown
assume Banach space Cauchy sequence chapter Cheeger-type comparison triangle constant map Corollary curvature upper bound define Definition denote differential doubling condition dx(x dx(y Ec(u EKS(u Ep(u Example exists follows function g gdfi geodesically complete gradient for ui harmonic maps Hence HKST homothetic Lemma liminf Lip u(z Lipu locally geodesics extendable locally Lipschitz continuous map between metric map between Riemannian map ui metric measure space metric space minimal generalized upper minimal geodesic Moreover obtain p-energy Proposition 3.3.6 Proposition 4.1.5 Rademacher's theorem respectively Riemannian manifold ru(p rx(x satisfies the doubling smooth Sobolev spaces space of curvature spaces for maps sub-embedding sufficiently small superrigidity Take a sequence Tohoku University totally geodesic map totally geodesic property triangle inequality uBr(zo unit speed curve upper gradient variation formula vector field weak Poincare inequality X,dx Zxyz