Spines and topology of thin Riemannian manifolds
Authors:
Stephanie B. Alexander and Richard L. Bishop
Journal:
Trans. Amer. Math. Soc. 355 (2003), 49334954
MSC (2000):
Primary 53C21, 57M50
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 3balls 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:
http://dx.doi.org/10.1090/S0002994703031635
PII:
S 00029947(03)031635
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
