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Download Lectures on Spaces of Nonpositive Curvature by Werner Ballmann PDF

a. This shows that a is nonwandering mod ~. 4. Hence ~ satisfies the duality condition. We come back to the beginning of this section. We said there that we will be in need of a large group of isometries, and we made this precise by introducing the duality condition.

8 implies ~ E r((). D e e e e e. 10 Remark. The duality condition has other useful technical properties. (1) Let Xl and X 2 be Hadamard spaces with metrics d 1 and d2 respectively and let X be the Hadamard space Xl x X 2 with the metric d = Jdi + d§. If r is a group of isometries of X satisfying the duality condition such that any 'P E r is of the form 'P = ('P1, 'P2), where 'Pi is an isometry of Xi, i = 1,2, then r 1 := {'P1 Ithere is an isometry 'P2 of X 2 with ('P17 'P2) E r} and the corresponding group r 2 satisfy the duality condition.

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