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Asymptotic invariants of smooth manifolds. (English. Russian original) Zbl 0812.57022
Russ. Acad. Sci., Izv., Math. 41, No. 1, 1-38 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 4, 707-751 (1992).
For a closed Riemannian manifold $$(M,g)$$, let $$\widetilde {M}$$ be some regular (infinite sheeted) covering and denote by $$\widetilde {g}$$ the metric of $$\widetilde {M}$$ obtained by lifting $$g$$. If $$q$$ is a point of $$\widetilde {M}$$ and $$V(t;q,g)$$ is the $$\widetilde{g}$$-geodesic ball on $$\widetilde{M}$$ with center $$q$$ and radius $$t$$, the function $$v(t;q,g) = \text{vol}_{\widetilde {g}} V(t;q,g)$$ reflects not only properties of the metric $$g$$ but also topological properties of $$M$$. The author defines and investigates certain new homotopy invariants of smooth manifolds, named absolute asymptotic volumes, which depend on the function $$v(t;q,g)$$ and are connected with various questions of the topology and geometry of manifolds; for example, the number of closed geodesics, the systolic constants and minimal surfaces.

##### MSC:
 57R99 Differential topology 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 57M10 Covering spaces and low-dimensional topology 49Q05 Minimal surfaces and optimization 52A40 Inequalities and extremum problems involving convexity in convex geometry 20F05 Generators, relations, and presentations of groups 53C22 Geodesics in global differential geometry 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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