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Obtuse constants of Alexandrov spaces. (English) Zbl 1443.53019
In [K. Grove and P. Petersen, Ann. Math. (2) 128, No. 1, 195–206 (1988; Zbl 0655.53032)], it is shown that there exist constants \(\delta\) and \(\epsilon\), depending on the dimension, curvature, volume and diameter of a Riemannian manifold, so that, if points \(p\) and \(q\) are \(\delta\)-close on the manifold, it is possible to draw a geodesic from one of the points which makes an obtuse angle of at least \(\frac{\pi}{2} + \epsilon\) with any shortest geodesic between \(p\) and \(q\). This same property holds true more generally in Alexandrov spaces.
The present paper asks whether, for a close pair \(p, q\), there is a third point \(x\) lying far from \(p\) and \(q\) with a shortest path from \(x\) to one of \(p\) or \(q\) that creates an obtuse angle with any shortest path between \(p\) and \(q\). We may take ‘far’ to mean ‘lying at least half the radius of \(M\) from the point where the angle is subtended’.
This is measured by a new geometric invariant. The obtuse constant of a pair gives a lower bound for the size of the obtuse angle which can be created by the best choice of \(x\). The obtuse constant of \(M\) itself is defined by taking the limit inferior as the distance between \(p\) and \(q\) approaches zero.
The obtuse constant is positive and is bounded below in terms of the dimension, curvature, volume and diameter, generalising the result of Grove and Petersen [loc. cit.]. The authors investigate its relationship to the volume, volume growth rate and asymptotic cone and also obtain rigidity results.
MSC:
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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References:
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