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Article

Keywords:
Obata equation; Tashiro equation; virtual Einstein equation
Summary:
In this paper, we prove new rigidity results related to some generalised Ricci-Hessian equation on Riemannian manifolds.
References:
[1] Alekseevskiĭ, D.V., Kimelfeld, B.N.: Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funktsional. Anal. i Prilozhen. 9 (2) (1975), 5–11. DOI 10.1007/BF01075445
[2] Besse, A.L.: Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete ed., vol. 3 (10), Springer, 1987. Zbl 0613.53001
[3] Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145 (1969), 1–49. DOI 10.1090/S0002-9947-1969-0251664-4
[4] Case, J.: The non-existence of quasi-Einstein metrics. Pacific J. Math. 248 (2) (2010), 277–284. DOI 10.2140/pjm.2010.248.277
[5] Case, J., Shu, Y.-J., Wei, G.: Rigidity of quasi-Einstein metrics. Differential Geom. Appl. 29 (2011), 93–100. DOI 10.1016/j.difgeo.2010.11.003
[6] Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Comm. Math. Phys. 214 (1) (2000), 137–189. DOI 10.1007/PL00005533
[7] Feitosa, F.E.S., Freitas Filho, A.A., Gomes, J.N.V., Pina, R.S.: Gradient almost Ricci soliton warped product. J. Geom. Phys. 143 (2019), 22–32. DOI 10.1016/j.geomphys.2019.05.003
[8] Ginoux, N., Habib, G., Kath, I.: A splitting theorem for Riemannian manifolds of generalised Ricci-Hessian type. hal-01877391, arXiv:1809.07546 (2018).
[9] Ginoux, N., Habib, G., Kath, I.: Skew Killing spinors in four dimensions. Ann. Global Anal. Geom. 59 (4) (2021), 501–535. DOI 10.1007/s10455-021-09754-9
[10] Güler, S., Demirbağ, S.A.: On warped product manifolds satisfying Ricci-Hessian class type equations. Publ. Inst. Math. (Beograd) (N.S.) 103 (117) (2018), 69–75. DOI 10.2298/PIM1817069G
[11] He, C., Petersen, P., Wylie, W.: Warped product Einstein metrics over spaces with constant scalar curvature. Asian J. Math. 18 (1) (2014), 159–189. DOI 10.4310/AJM.2014.v18.n1.a9
[12] He, C., Petersen, P., Wylie, W.: Uniqueness of warped product Einstein metrics and applications. J. Geom. Anal. 25 (4) (2015), 2617–2644. DOI 10.1007/s12220-014-9528-8
[13] He, C., Petersen, P., Wylie, W.: Warped product Einstein metrics on homogeneous spaces and homogeneous Ricci solitons. J. Reine Angew. Math. 707 (2015), 217–245. DOI 10.1515/crelle-2013-0078
[14] He, C., Petersen, P., Wylie, W.: Warped product rigidity. Asian J. Math. 19 (1) (2015), 135–170. DOI 10.4310/AJM.2015.v19.n1.a6
[15] Kanai, M.: On a differential equation characterizing a Riemannian structure of a manifold. Tokyo J. Math. 6 (1) (1983), 143–151. DOI 10.3836/tjm/1270214332
[16] Karp, L.: Subharmonic functions, harmonic mappings and isometric immersions. Seminar on Differential Geometry, Ann. of Math. Stud 102 (1982), 133–142.
[17] Karp, L.: Subharmonic functions on real and complex manifolds. Math. Z. 179 (4) (1982), 535–554. DOI 10.1007/BF01215065
[18] Kim, D.-S., Kim, Y.H.: Compact Einstein warped product spaces with nonpositive scalar curvature. Proc. Amer. Math. Soc. 131 (8) (2003), 2573–2576. DOI 10.1090/S0002-9939-03-06878-3
[19] Kühnel, W., Rademacher, H.-B.: Conformally Einstein product spaces. Differential Geom. Appl. 49 (2016), 65–96. DOI 10.1016/j.difgeo.2016.07.005
[20] Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14 (1962), 333–340. DOI 10.2969/jmsj/01430333
[21] Petersen, P., Wylie, W.: Rigidity of homogeneous gradient soliton metrics and related equations. Differential Geom. Appl. 84 (2022), no. 101929, 29. DOI 10.1016/j.difgeo.2022.101929
[22] Ranjan, A., Santhanam, G.: A generalization of Obata’s theorem. J. Geom. Anal. 7 (3) (1997), 357–375. DOI 10.1007/BF02921625
[23] Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117 (1965), 251–275. DOI 10.1090/S0002-9947-1965-0174022-6
[24] Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28 (1975), 201–228. DOI 10.1002/cpa.3160280203
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