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Critical points of distance functions and applications to geometry. (English) Zbl 0771.53015
Geometric topology: recent developments, Lect. 1st Sess. CIME, Montecatini Terme/Italy 1990, Lect. Notes Math. 1504, 1-38 (1991).
[For the entire collection see Zbl 0746.00065.]
This article is the text of a set of lecture notes given by the author for a C.I.M.E. short course at Montecatini Terme in June 1990. The author discusses three basic results in Riemannian geometry that are proved by combining the notion of a critical point of a distance function with the Toponogov comparison theorem for geodesic triangles. This technique was introduced by K. Grove and K. Shiohama.
The three results discussed (with proofs) are: 1) the Grove-Petersen theorem of the finiteness of homotopy types of manifolds admitting metrics with suitable bounds on diameter, volume and curvature; 2) Gromov’s bound on the Betti numbers in terms of curvature and diameter; 3) the Abresch-Gromoll theorem on finiteness of topological type for manifolds with nonnegative Ricci curvature, sectional curvature bounded below and slow diameter growth. We omit precise statements of these results. We note that the author proved the first (diffeomorphism) version of 1) (see Theorem 3.1 of the text) and pioneered the subject of finiteness theorems for compact Riemannian manifolds with bounded geometry.
Let \(p\) be a point in a Riemannian manifold \(M\), and let \(\rho_ p: M \to \mathbb{R}\) denote the distance function given by \(\rho_ p(q) = d(p,q)\) for \(q \in M\). This function is smooth in \(M-\{C(p) \cup p\}\), where \(C(p)\) denotes the cutlocus of \(p\), but \(\rho_ p\) is not smooth on all of \(M\). A point \(q\) of \(M\) is said to be a critical point of \(\rho_ p\) if for every nonzero vector \(v \in T_ qM\) there exists a minimal geodesic \(\gamma\) from \(q = \gamma(0)\) to \(p = \gamma(1)\) such that the angle between \(v\) and \(\gamma'(0)\) is at most \(\pi/2\). In particular, if \(\rho_ p\) has a maximum value at \(q\), then \(q\) is a critical point of \(\rho_ p\), a fact which M. Berger observed earlier. Using this definition for a critical point of a distance function many of the techniques of Morse theory and Lusternik-Schnirelman theory carry over to this nonsmooth situation. For further details see the article “Critical point theory for distance functions” by K. Grove (preprint 1991).
The text here includes many examples as well as proofs of the major results. Included is a detailed treatment of the Bishop-Gromov inequalities and later refinements, which relate lower bounds on the Ricci curvature of a manifold \(M\) to upper bounds on the volumes of metric balls or more generally volumes of tubular neighborhoods \(T_ r(X)\), where \(X\) is a compact subset of \(M\) and \(r\) is the radius of the neighborhood. An important role in the discussion is played by the Laplacian of the distance function \(\rho_ p\), and by the maximum principle that holds for functions \(f\) that satisfy the inequality \(\Delta f \geq 0\) in the barrier sense.

53C20 Global Riemannian geometry, including pinching