DOI QR코드

DOI QR Code

양의 단면 곡률을 가지는 컴팩트 공간에 대하여

  • 고관석 (인하대학교 이과대학 수학통계학부)
  • 발행 : 2005.04.01

초록

리만 기하학에서 중요한 문제중의 하나는 주어진 곡률부호를 가지는 다양체를 분류하는 것이다. 그렇게 하기 위해서는 곡률과 위상과의 상호 관계를 밝히는 것이 중요하다. 특히 양의 곡률을 가지는 공간을 분류하는 것은 어려운 문제로 알려져 있으며 위상적 성질에 대해서도 알려진 것은 매우 적다. 본 논문에서는 지금까지 알려진 양의 곡률을 가지는 공간들을 살펴 보고 그들 공간들에 대한 일반적인 정리들과 호프의 문제를 소개하고자 한다.

참고문헌

  1. S. Aloff and N. Wallach, An infinite family of distinct 7-manifolds admitting positively curved manifolds, Bull. Amer. math. Soc. 81 (1975), 93-97 https://doi.org/10.1090/S0002-9904-1975-13649-4
  2. A. Bazaikin, On a family of 13-dimensional closed Riemannian manifolds of positive secional curvature, Thesis, Univ. of Novosibirisk, 1995
  3. M. Berger, Les varietes riemanniennes 1/4-pincees , Ann. Scuola Norm. Sup. Pisa 14 (1960), 161-170
  4. M. Berger, Les varietes riemanniennes homogenes normales simplement connexes a coubure stictement positive , Ann. Scuola Norm. Sup. Pisa 15 (1961), 179-246
  5. A. Besse, Einstein Manifolds, Springer-Verlag, 1987
  6. S. Bochner, Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776-797 https://doi.org/10.1090/S0002-9904-1946-08647-4
  7. J. Cheeger , Some examples of manifolds of nonnegative Curvature, J. Differential Geom. 8 (1973), 623-628 https://doi.org/10.4310/jdg/1214431964
  8. J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North Holland, Amsterdam, 1975
  9. S. S. Chern , On curvature and Charateristic Class of a Riemannian Manifold, J. Abh. Math. Sem. Univ. Hamhurg 20 (1955), 117-126 https://doi.org/10.1007/BF02960745
  10. S. S. Chern, The Geometry of G-structure, Bull. Amer. Math. Soc. 72 (1966), 167-219
  11. J. H. Eschenberg, Inhomogeneous space of positive curvature, Differential Geom. Appl. 2 (1992), 123-132 https://doi.org/10.1016/0926-2245(92)90029-M
  12. J. H. Eschenberg, New examples of manifolds with stictly positive curvature, Invent. Math. 66 (1982), 469-480 https://doi.org/10.1007/BF01389224
  13. R. Geroch, Positive sectional curvature does not imply positive Gauss-Bonnet integrand, Proc. Amer. Math. Soc. 54 (1976), 267-270 https://doi.org/10.2307/2040798
  14. D. Gromoll and W. T. Meyer, An exotic sphere with nonnegative sectional cur- vature, Ann. of Math. 100 (1974), 401-406 https://doi.org/10.2307/1971078
  15. R. Hamilton, 3-manifolds with positive Ricci curvature, J. Differential Geom. 12 (1982), 255-306
  16. R. Hamilton, Formation of singularities in the Ricci flow, Surv. Differ. Geom. 2 (1985), 7-136
  17. R. Hamilton, Non-singular solutions of the Ricci flow on 3-manifolds, Comm. Anal. Geom. 7 (1999), 695-729 https://doi.org/10.4310/CAG.1999.v7.n4.a2
  18. H. Hopf, Differentilageometrie und topologische Gestalt, Jahres-berichtd. DMV. 41 (1932), 209-229
  19. S. B. Myers, Riemannian Manifolds with Positive Mean Curvature, Duke Math. J. 8 (1941), 401-404 https://doi.org/10.1215/S0012-7094-41-00832-3
  20. B. O'neill, The fundamental equation of a submerion, Michigan Math. J. 23 (1966), 459-469
  21. H. Samelson, On curvature and characteristic of homogenous spaces, Michigan Math. J. 5 (1958), 13-18 https://doi.org/10.1307/mmj/1028998006
  22. J. L. Synge, On the Connectivity of Spaces of Positive Curvature, Q. J. Math. 7 (1936), 316-320 https://doi.org/10.1093/qmath/os-7.1.316
  23. S. Tanno, Pomenades on spheres, Tokyo Ibst. Tech. 1996
  24. N. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. 96 (1972), 277-295 https://doi.org/10.2307/1970789
  25. S. T. Yau, Problem section , Seminar on Differential Geometry, Princeton Univ. Press, 1977, 669-709
  26. F. Zheng, Complex Differential Geometry, Studies in Advanced Math. 18 (2000)