Abstract
In this paper we establish two local versions of the Cheeger-Gromoll Splitting Theorem. We show that if a complete Riemannian manifold M has nonnegative Ricci curvature outside a compact set B and contains a line γ which does not intersect B, then the line splits in a maximal neighborhood that is contained in M \ B. We use this result to give a simplified proof that M has a bounded number of ends. We also prove that if M has sectional curvature which is nonnegative (and bounded from above) in a tubular neighborhood U of a geodesic γ which is a line in U, then U splits along γ.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1231-1239 |
| Number of pages | 9 |
| Journal | Proceedings of the American Mathematical Society |
| Volume | 120 |
| Issue number | 4 |
| DOIs | |
| State | Published - 1994 |
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Keywords
- Asymptotic ray
- Line
- Ricci curvature
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
Local splitting theorems for riemannian manifolds. / Cai, MingLiang; Galloway, Gregory J; Liu, Z.
In: Proceedings of the American Mathematical Society, Vol. 120, No. 4, 1994, p. 1231-1239.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Local splitting theorems for riemannian manifolds
AU - Cai, MingLiang
AU - Galloway, Gregory J
AU - Liu, Z.
PY - 1994
Y1 - 1994
N2 - In this paper we establish two local versions of the Cheeger-Gromoll Splitting Theorem. We show that if a complete Riemannian manifold M has nonnegative Ricci curvature outside a compact set B and contains a line γ which does not intersect B, then the line splits in a maximal neighborhood that is contained in M \ B. We use this result to give a simplified proof that M has a bounded number of ends. We also prove that if M has sectional curvature which is nonnegative (and bounded from above) in a tubular neighborhood U of a geodesic γ which is a line in U, then U splits along γ.
AB - In this paper we establish two local versions of the Cheeger-Gromoll Splitting Theorem. We show that if a complete Riemannian manifold M has nonnegative Ricci curvature outside a compact set B and contains a line γ which does not intersect B, then the line splits in a maximal neighborhood that is contained in M \ B. We use this result to give a simplified proof that M has a bounded number of ends. We also prove that if M has sectional curvature which is nonnegative (and bounded from above) in a tubular neighborhood U of a geodesic γ which is a line in U, then U splits along γ.
KW - Asymptotic ray
KW - Line
KW - Ricci curvature
UR - http://www.scopus.com/inward/record.url?scp=84966237738&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84966237738&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-1994-1186984-2
DO - 10.1090/S0002-9939-1994-1186984-2
M3 - Article
AN - SCOPUS:84966237738
VL - 120
SP - 1231
EP - 1239
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
SN - 0002-9939
IS - 4
ER -