Spines and topology of thin Riemannian manifolds

Authors:
Stephanie B. Alexander and Richard L. Bishop

Journal:
Trans. Amer. Math. Soc. **355** (2003), 4933-4954

MSC (2000):
Primary 53C21, 57M50

DOI:
https://doi.org/10.1090/S0002-9947-03-03163-5

Published electronically:
July 28, 2003

MathSciNet review:
1997598

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Abstract: Consider Riemannian manifolds for which the sectional curvature of and second fundamental form of the boundary are bounded above by one in absolute value. Previously we proved that if has sufficiently small inradius (i.e. all points are sufficiently close to the boundary), then the cut locus of exhibits canonical branching behavior of arbitrarily low branching number. In particular, if is *thin* in the sense that its inradius is less than a certain universal constant (known to lie between and ), then collapses to a triply branched simple polyhedral spine.

We use a graphical representation of the stratification structure of such a collapse, and relate numerical invariants of the graph to topological invariants of when is simply connected. In particular, the number of connected strata of the cut locus is a topological invariant. When is -dimensional and compact, has complexity in the sense of Matveev, and is a connected sum of copies of the real projective space , copies chosen from the lens spaces , and handles chosen from or , with 3-balls removed, where . Moreover, we construct a thin metric for every graph, and hence for every homeomorphism type on the list.

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Additional Information

**Stephanie B. Alexander**

Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, Illinois 61801

Email:
sba@math.uiuc.edu

**Richard L. Bishop**

Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, Illinois 61801

Email:
bishop@math.uiuc.edu

DOI:
https://doi.org/10.1090/S0002-9947-03-03163-5

Keywords:
Riemannian manifolds with boundary,
collapse,
$3$-manifolds,
curvature bounds,
inradius,
stratification

Received by editor(s):
June 4, 2001

Received by editor(s) in revised form:
July 12, 2002

Published electronically:
July 28, 2003

Article copyright:
© Copyright 2003
American Mathematical Society