Asymptotic invariants of smooth manifolds. (English. Russian original) Zbl 0812.57022

Babenko, I. K.

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.

Reviewer: A.Cavicchioli (Modena)

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.)

Keywords:

homotopy invariants; absolute asymptotic volumes; closed geodesics; systolic constants; minimal surfaces