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In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.
Shoshichi Kobayashi was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie algebras.
In mathematics, Salomon Bochner proved in 1946 that any Killing vector field of a compact Riemannian manifold with negative Ricci curvature must be zero. Consequently the isometry group of the manifold must be finite.
References
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- Kobayashi, S.; Nomizu, K. (2009) [1969]. Foundations of Differential Geometry. Wiley Classics Library. Vol. 2. Wiley. ISBN 978-0-471-15732-8. Zbl 0175.48504.
- Eells, J. (1963). "Review of Volume 1" (PDF). Mathematical Reviews .
- Hermann, Robert (1964). "Review: Foundations of differential geometry, Volume 1". Bulletin of the American Mathematical Society . 70 (2): 232–235. doi: 10.1090/s0002-9904-1964-11094-6 .
- Eells, J. (1969). "Review of Volume 2" (PDF). Mathematical Reviews.