A New Differentiable Sphere Theorem and Its Applications

Year 2024, Volume: 17 Issue: 2, 388 - 393, 27.10.2024

Vladimir Rovenski , Sergey Stepanov

https://doi.org/10.36890/iejg.1529961

Abstract

In this paper, we use the Lichnerowicz Laplacian to prove new results: the sphere theorem and the integral inequality for Einstein's infinitesimal deformations, which allow us to characterize spherical space forms. Our version of the sphere theorem states that a closed connected Riemannian manifold $(M, g)$ of even dimension $n>3$ is diffeomorphic to a Euclidean sphere or a real projective space if the inequality $Ric_{\rm max}(x) < n K_{\rm min}(x) g$ is true at each point $x\in M$, where $Ric_{\rm max}(x)$ is the maximum of the Ricci curvature, and $K_{\rm min}(x)$ is the minimum of the sectional curvature of $(M, g)$ at $x$. Since this inequality implies positive sectional curvature; therefore, our result partially answers Hopf's old open question.

Keywords

Lichnerowicz Laplacian, differentiable sphere theorem, Einstein's infinitesimal deformation, spherical space form, curvature operator of the second kind

Ethical Statement

Not required

Supporting Institution

No

Thanks

Not required

References

• [1] Berger, M., Ebin, D.: Some decomposition of the space of symmetric tensors of a Riemannian manifold. Journal of Diff. Geometry, 3, 379-392 (1969).
• [2] Besse, A.L.: Einstein Manifolds, Berlin, Springer (1987).
• [3] Bishop, R.L., Goldberg, S.I.: Some implications of the generalized Gauss-Bonnet theorem. Transactions of the AMS 112 (3), 508-535 (1964).
• [4] Brendle, S., Schoen, R.M.: Classification of manifolds with weakly 1/4-pinched curvatures. Acta Math. 200, 1-13 (2008).
• [5] Brendle, S., Schoen, R.: Curvature, sphere theorems, and the Ricci flow. Bull. Amer. Math. Soc. (N.S.) 48 (1), 1-32 (2011).
• [6] Cao, X., Gursky, M.J., Tran, H.: Curvature of the second kind and a conjecture of Nishikawa. Commentarii Mathematici Helvetici, 98 (1), 195-216 (2023).
• [7] Chern, S.S.: On the curvature and characteristic classes of a Riemannian manifold. Abh. Math. Sem. Univ. Hamburg, 20, 117-126 (1956).
• [8] Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow, AMS Graduate Studies in Mathematics, 77, Providence, RI (2006).
• [9] Lichnerowicz, A.: Propagateurs et commutateurs en relativite generate. Publ. Math., Inst. Hautes Étud. Sci. 10, 293-344 (1961).
• [10] Mikeš, J., Rovenski, V., Stepanov, S.: An example of Lichnerowicz-type Laplacian. Ann. Global Anal. Geom. 58 (1), 19-34 (2020).
• [11] Mikes, J., Rovenski, V., Stepanov, S., Tsyganok, I.: Application of the generalized Bochner technique to the study of conformally flat Riemannian manifolds. Mathematics, 9, 927 (2021).
• [12] Rovenski, V., Stepanov, S., Tsyganok, I.: On the Betti and Tachibana numbers of compact Einstein manifolds. Mathematics, 7, 1210 (2019).
• [13] Tachibana, Sh., Ogiue, K.: Les variétés riemanniennes dont l’opérateur de coubure restreint est positif sont des sphéres d’homologie réelle. C. R. Acad. Sci. Paris, 289, 29-30 (1979).
• [14] Wolf, J.: Spaces of constant curvature, Publish or Perish, Houston TX (1984).
• [15] Xu, H.-W., Gu, J.-Ru.: The differentiable sphere theorem for manifolds with positive Ricci curvature, Proc. AMS 140 (3), 1011-1021 (2012).
• [16] Yano, K., Bochner, S.: Curvature and Betti numbers, Princeton, N. J., Princeton University Press (1953).