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References 1. Böhm, C.; Wilking, B. - Manifolds with positive curvature operators are space forms, Ann. of Math., 167 (2008), 1079-1097. 2. Coste, M. - An Introduction to Semialgebraic Geometry, Universita di Pisa, Dipartimento di Matematica, 2000, available at http://perso.univ-rennesl.fr/michel.coste/polyens/SAG.pdf 3. Gilbert, G.T. - Positive definite matrices and Sylvester’s criterion, Amer. Math. Monthly, 98 (1991), 44-46. 4. Gallot, S.; Hulin, D.; Lafontaine, J. - Riemannian Geometry, Third edition, Universitext, Springer-Verlag, Berlin, 2004. 5. Kobayashi

References 1. Cao, X. - Compact gradient shrinking Ricci solitons with positive curvature operator, J. Geom. Anal., 17 (2007), 425-433. 2. Chow, B.; Lu, P.; Ni, L. - Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77, American Mathematical Society, Providence, RI; Science Press, New York, 2006. 3. Fischer, A.E. - An introduction to conformal Ricci flow, http://arxiv.org/abs/math/0312519, 2003, 1-52. 4. Hamilton, R.S. - Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306. 5. M¨uller, R. - Differential Harnack

Abstract

In this paper, the standard almost complex structure on the tangent bunle of a Riemannian manifold will be generalized. We will generalize the standard one to the new ones such that the induced (0, 2)-tensor on the tangent bundle using these structures and Liouville 1-form will be a Riemannian metric. Moreover, under the integrability condition, the curvature operator of the base manifold will be classified.

­temporary Mathematics, 3, Science Press and Graduate Studies in Math­ematics, 77, American Mathematical Society (co-publication), 2006. [5] R. S. Hamilton, Four manifolds with positive curvature operator, J. Diff. Geom., 24, (1986), 153-179. [6] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159

Ricci soliton, Analele Stiintifice Ale Universitatii Al. I. Cuza Din Iasi(S.N) Mathematica, Tomul LXI, 61, (2015), 245-252. [5] C. Calin and M. Crasmareanu, From the Eisenhart problem to the Ricci solitons in f-Kenmotsu manifolds, Bull. Malay. Math. Soc., 33, (2010), 361-368. [6] X. Cao, Compact gradient shrinking Ricci solitons with positive curvature operator, J. Geom. Anal., 17, (2007), 425-433. [7] B. Chow, P. Lu, and L. Ni, Hamilton’s Ricci ow, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI; Science Press, New York, 77, (2006) [8

. Differential Geom. , 17/2 , (1982), 256–306 [26] R. Hamilton , Four-manifolds with positive curvature operator, J. Differential Geom. , 24/2 , (1986), 153–179 [27] R. S. Hamilton , The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math., American Math. Soc., Providence, RI , 71 , (1988), 237–262 [28] R. Sharma , Second order parallel tensor in real and complex space forms, International Journal of Mathematics and Mathematical Sciences , 12/4 , (1989), 787–790 [29] R. Sharma , Second order parallel tensors on contact

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