## Euler-Poincaré characteristic and higher order sectional curvature. I

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- by Chuan-Chih Hsiung and Kenneth Michael Shiskowski PDF
- Trans. Amer. Math. Soc.
**305**(1988), 113-128 Request permission

## Abstract:

The following long-standing conjecture of H. Hopf is well known. Let $M$ be a compact orientable Riemannian manifold of even dimension $n \geqslant 2$. If $M$ has nonnegative sectional curvature, then the Euler-Poincaré characteristic $\chi (M)$ is nonnegative. If $M$ has nonpositive sectional curvature, then $\chi (M)$ is nonnegative or nonpositive according as $n \equiv 0$ or $2\bmod 4$. This conjecture for $n = 4$ was proved first by J. W. Milnor and then by S. S. Chern by a different method. The main object of this paper is to prove this conjecture for a general $n$ under an extra condition on higher order sectional curvature, which holds automatically for $n = 4$. Similar results are obtained for Kähler manifolds by using holomorphic sectional curvature, and F. Schur’s theorem about the constancy of sectional curvature on a Riemannian manifold is extended.## References

- R. L. Bishop and S. I. Goldberg,
*Some implications of the generalized Gauss-Bonnet theorem*, Trans. Amer. Math. Soc.**112**(1964), 508–535. MR**163271**, DOI 10.1090/S0002-9947-1964-0163271-8 - R. L. Bishop and S. I. Goldberg,
*On the second cohomology group of a Kaehler manifold of positive curvature*, Proc. Amer. Math. Soc.**16**(1965), 119–122. MR**172221**, DOI 10.1090/S0002-9939-1965-0172221-6 - Spencer Bloch and David Gieseker,
*The positivity of the Chern classes of an ample vector bundle*, Invent. Math.**12**(1971), 112–117. MR**297773**, DOI 10.1007/BF01404655 - Jean-Pierre Bourguignon and Hermann Karcher,
*Curvature operators: pinching estimates and geometric examples*, Ann. Sci. École Norm. Sup. (4)**11**(1978), no. 1, 71–92. MR**493867**, DOI 10.24033/asens.1340 - Shiing-shen Chern,
*On curvature and characteristic classes of a Riemann manifold*, Abh. Math. Sem. Univ. Hamburg**20**(1955), 117–126. MR**75647**, DOI 10.1007/BF02960745 - Yuk-keung Cheung and Chuan-chih Hsiung,
*Curvature and characteristic classes of compact Riemannian manifolds*, J. Differential Geometry**1**(1967), no. 1, 89–97. MR**217738** - Robert Geroch,
*Positive sectional curvatures does not imply positive Gauss-Bonnet integrand*, Proc. Amer. Math. Soc.**54**(1976), 267–270. MR**390961**, DOI 10.1090/S0002-9939-1976-0390961-8 - Alfred Gray,
*A generalization of F. Schur’s theorem*, J. Math. Soc. Japan**21**(1969), 454–457. MR**248681**, DOI 10.2969/jmsj/02130454 - Alfred Gray,
*Some relations between curvature and characteristic classes*, Math. Ann.**184**(1969/70), 257–267. MR**261492**, DOI 10.1007/BF01350854 - Alfred Gray,
*Chern numbers and curvature*, Amer. J. Math.**100**(1978), no. 3, 463–476. MR**493905**, DOI 10.2307/2373833
W. V. Hodge and D. Pedoe, - David L. Johnson,
*Curvature and Euler characteristic for six-dimensional Kähler manifolds*, Illinois J. Math.**28**(1984), no. 4, 654–675. MR**761996** - Paul F. Klembeck,
*On Geroch’s counterexample to the algebraic Hopf conjecture*, Proc. Amer. Math. Soc.**59**(1976), no. 2, 334–336. MR**428225**, DOI 10.1090/S0002-9939-1976-0428225-6 - Shoshichi Kobayashi and Katsumi Nomizu,
*Foundations of differential geometry. Vol. I*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR**1393940** - Ravindra S. Kulkarni,
*On the Bianchi Identities*, Math. Ann.**199**(1972), 175–204. MR**339004**, DOI 10.1007/BF01429873 - Daniel Meyer,
*Sur les variétés riemanniennes à opérateur de courbure positif*, C. R. Acad. Sci. Paris Sér. A-B**272**(1971), A482–A485 (French). MR**279736** - Shigefumi Mori,
*Projective manifolds with ample tangent bundles*, Ann. of Math. (2)**110**(1979), no. 3, 593–606. MR**554387**, DOI 10.2307/1971241 - A. M. Naveira,
*Caractérisation des variétés à courbures sectionnelles holomorphes généralisées constantes*, J. Differential Geometry**9**(1974), 55–60 (French). MR**417981**, DOI 10.4310/jdg/1214432090 - A. M. Naveira,
*On the higher order sectional curvatures*, Illinois J. Math.**19**(1975), 165–172. MR**377752**, DOI 10.1215/ijm/1256050808 - I. M. Singer and J. A. Thorpe,
*The curvature of $4$-dimensional Einstein spaces*, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 355–365. MR**0256303** - Yum Tong Siu and Shing Tung Yau,
*Compact Kähler manifolds of positive bisectional curvature*, Invent. Math.**59**(1980), no. 2, 189–204. MR**577360**, DOI 10.1007/BF01390043 - Ann Stehney,
*Courbure d’ordre $p$ et les classes de Pontrjagin*, J. Differential Geometry**8**(1973), 125–134 (French). MR**362333** - Ann Stehney,
*Extremal sets of $p$-th sectional curvature*, J. Differential Geometry**8**(1973), 383–400. MR**341343** - John A. Thorpe,
*Sectional curvatures and characteristic classes*, Ann. of Math. (2)**80**(1964), 429–443. MR**170308**, DOI 10.2307/1970657 - John A. Thorpe,
*On the curvatures of Riemannian manifolds*, Illinois J. Math.**10**(1966), 412–417. MR**196681** - John A. Thorpe,
*Some remarks on the Gauss-Bonnet integral*, J. Math. Mech.**18**(1969), 779–786. MR**0256307**
—,

*Methods of algebraic geometry*, Cambridge Univ. Press, Cambridge, 1947.

*On the curvature tensor of a positively curved*$4$-

*manifold*, Proc. 13 th Biennial Sem. Canad. Math. Congress, Vol. 2, 1971, pp. 156-159.

## Additional Information

- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**305**(1988), 113-128 - MSC: Primary 53C20; Secondary 53C55, 57R20
- DOI: https://doi.org/10.1090/S0002-9947-1988-0920149-3
- MathSciNet review: 920149