Sectional curvature

Spectral geometry

Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018

Since Sm is simply connected, τ defines an isomorphism between G and π1(Mτ). The manifold Mτ inherits a Riemannian metric of constant sectional curvature +1; every compact Riemannian manifold with such a metric arises in this way. These are the spherical space forms and they have been classified; see Wolf.

Instability of planetary flows using Riemann curvature: a numerical study

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Journal Information

Published in Geophysical & Astrophysical Fluid Dynamics, 2019

Richard Blender

The aim of the present work is to derive numerical results for the sectional curvature of nondivergent zonal flows on the sphere. As in the publications cited above we identify negative sectional curvature with instability. The calculations are based on Arakelyan and Savvidy (1989) and Dowker and Mo-Zheng (1990). The results are visualised for zonal flows with low total wave numbers which correspond to observations. The planetary rotation is considered by adding a solid body rotation represented by the stream-function to the zonal flow stream-functions. This paper is organised as follows: In section 2 the instability of spherical flows is described in terms of the sectional curvature which is based on the Riemann curvature tensor. In section 3 the results are presented and in section 4 the findings are summarised and discussed.

Sectional curvature

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Spectral geometry

Instability of planetary flows using Riemann curvature: a numerical study