Scalar curvature estimates by parallel alternating torsion
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- by Sebastian Goette
- Trans. Amer. Math. Soc. 363 (2011), 165-183
- DOI: https://doi.org/10.1090/S0002-9947-2010-04878-0
- Published electronically: August 20, 2010
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Abstract:
We generalize Llarull’s scalar curvature comparison to Riemannian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on $\Lambda ^2TM$. As a by-product, we show that the Euler number and signature of such manifolds are determined by their global holonomy representation. Our result holds in particular for all quotients of compact Lie groups of equal rank, equipped with a normal homogeneous metric.
We also correct a mistake in the treatment of odd-dimensional spaces in our earlier papers.
References
- Ilka Agricola and Thomas Friedrich, On the holonomy of connections with skew-symmetric torsion, Math. Ann. 328 (2004), no. 4, 711–748. MR 2047649, DOI 10.1007/s00208-003-0507-9
- B. Ammann (joint with M. Dahl, E. Humbert), A surgery formula for the smooth Yamabe invariant, in: J. Brüning, R. Mazzeo, P. Piazza (eds.), Analysis and Geometric Singularities, Oberwolfach Reports 4 (2007), 2413–2417.
- B. Ammann, M. Dahl, E. Humbert, Smooth Yamabe invariant and surgery, preprint (2008); arXiv:0804.1418.
- Christian Bär, On nodal sets for Dirac and Laplace operators, Comm. Math. Phys. 188 (1997), no. 3, 709–721. MR 1473317, DOI 10.1007/s002200050184
- Christoph Böhm and Burkhard Wilking, Manifolds with positive curvature operators are space forms, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 683–690. MR 2275617
- Huai Dong Cao and Bennett Chow, Compact Kähler manifolds with nonnegative curvature operator, Invent. Math. 83 (1986), no. 3, 553–556. MR 827367, DOI 10.1007/BF01394422
- Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
- Richard Cleyton and Andrew Swann, Einstein metrics via intrinsic or parallel torsion, Math. Z. 247 (2004), no. 3, 513–528. MR 2114426, DOI 10.1007/s00209-003-0616-x
- S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. (9) 54 (1975), no. 3, 259–284 (French). MR 454884
- S. Goette, Äquivariante $\eta$-Invarianten homogener Räume, Shaker, Aachen (1997).
- Sebastian Goette, Equivariant $\eta$-invariants on homogeneous spaces, Math. Z. 232 (1999), no. 1, 1–42. MR 1714278, DOI 10.1007/PL00004757
- Sebastian Goette, Vafa-Witten estimates for compact symmetric spaces, Comm. Math. Phys. 271 (2007), no. 3, 839–851. MR 2291798, DOI 10.1007/s00220-007-0188-4
- S. Goette and U. Semmelmann, $\textrm {Spin}^c$ structures and scalar curvature estimates, Ann. Global Anal. Geom. 20 (2001), no. 4, 301–324. MR 1876863, DOI 10.1023/A:1013035721335
- S. Goette and U. Semmelmann, Scalar curvature estimates for compact symmetric spaces, Differential Geom. Appl. 16 (2002), no. 1, 65–78. MR 1877585, DOI 10.1016/S0926-2245(01)00068-7
- M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993) Progr. Math., vol. 132, Birkhäuser Boston, Boston, MA, 1996, pp. 1–213. MR 1389019, DOI 10.1007/s10107-010-0354-x
- V. F. Kiričenko, $K$-spaces of maximal rank, Mat. Zametki 22 (1977), no. 4, 465–476 (Russian). MR 474103
- H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
- M. Listing, Scalar curvature on compact symmetric spaces, Preprint.
- Marcelo Llarull, Scalar curvature estimates for $(n+4k)$-dimensional manifolds, Differential Geom. Appl. 6 (1996), no. 4, 321–326. MR 1422338, DOI 10.1016/S0926-2245(96)00025-3
- Marcelo Llarull, Sharp estimates and the Dirac operator, Math. Ann. 310 (1998), no. 1, 55–71. MR 1600027, DOI 10.1007/s002080050136
- Paul-Andi Nagy, Nearly Kähler geometry and Riemannian foliations, Asian J. Math. 6 (2002), no. 3, 481–504. MR 1946344, DOI 10.4310/AJM.2002.v6.n3.a5
- R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. MR 318398, DOI 10.2307/1970892
- Shun-ichi Tachibana, A theorem on Riemannian manifolds of positive curvature operator, Proc. Japan Acad. 50 (1974), 301–302. MR 365415
- N. Weinert, Twistorräume von Quaternionisch-Kähler-Mannigfaltigkeiten und Skalarkrümmung, diploma thesis Uni. Freiburg, 2010.
Bibliographic Information
- Sebastian Goette
- Affiliation: Mathematisches Institut, Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany
- Email: sebastian.goette@math.uni-freiburg.de
- Received by editor(s): October 10, 2007
- Received by editor(s) in revised form: July 25, 2008
- Published electronically: August 20, 2010
- Additional Notes: This research was supported in part by DFG special programme “Global Differential Geometry”
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 165-183
- MSC (2010): Primary 53C21; Secondary 58J20, 53C15, 53C30
- DOI: https://doi.org/10.1090/S0002-9947-2010-04878-0
- MathSciNet review: 2719677