Let be a noncompact complete Riemannian manifold of dimension n, and let be an integrable subbundle of TM. Let be the restricted metric on F and let be the associated leafwise scalar curvature. Let be a smooth area decreasing map along F, which is locally constant near infinity and of non-zero degree. We show that if on the support of , and either TM or F is spin, then . As a consequence, we prove Gromov’s sharp foliated -twisting conjecture. Using the same method, we also extend two famous non-existence results due to Gromov and Lawson about -enlargeable metrics (and/or manifolds) to the foliated case.
Funding statement: Guangxiang Su and Weiping Zhang were partially supported by NSFC Grant No. 11931007 and Nankai Zhide Foundation. Xiangsheng Wang was partially supported by NSFC Grant No. 12101361, the project of Young Scholars of SDU and the fundamental research funds of Shandong University, Grant No. 2020GN063.
The authors would like to thank the anonymous referee for careful reading and valuable suggestions.
 A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras (Kyoto 1983), Pitman Res. Notes Math. Ser. 123, Longman Scientific & Technical, Harlow (1986), 52–144. Search in Google Scholar
 M. Gromov and H. B. Lawson, Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. Inst. Hautes Études Sci. 58 (1983), 83–196. 10.1007/BF02953774Search in Google Scholar
 H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, Princeton Math. Ser. 38, Princeton University, Princeton 1989. Search in Google Scholar
 A. Lichnerowicz, Spineurs harmoniques, C. R. Math. Acad. Sci. Paris 257 (1963), 7–9. Search in Google Scholar
 K. Liu and W. Zhang, Adiabatic limits and foliations, Topology, geometry, and algebra: Interactions and new directions (Stanford 1999), Contemp. Math. 279, American Mathematical Society, Providence (2001), 195–208. 10.1090/conm/279/04561Search in Google Scholar
 W. Zhang, Positive scalar curvature on foliations: The enlargeability, Geometric analysis—in honor of Gang Tian’s 60th birthday, Progr. Math. 333, Birkhäuser/Springer, Cham (2020), 537–544. 10.1007/978-3-030-34953-0_22Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston