Abstract
The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold M can give information about the stability of inviscid, incompressible fluid flows on M. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by \(\mathcal {D}_{\mu ,E}(M)\), has positive sectional curvature in every section containing the field \(X = u(r)\partial _\theta \) iff \(\partial _r(ru^2)>0\). This is in sharp contrast to the situation on \(\mathcal {D}_{\mu }(M)\), where only Killing fields X have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on \(\mathcal {D}_{\mu ,E}(M)\) along the geodesic defined by X.
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Notes
This was proved rigorously by Ebin and Marsden (1970), by working in the context of Sobolev \(H^s\) diffeomorphisms for \(s>\tfrac{1}{2}\dim {M}+1\). Here for simplicity we will work in the context of smooth diffeomorphisms since the curvature formulas are the same either way.
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The second author gratefully acknowledges support from NSF Grants DMS-1157293 and DMS-1105660.
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Washabaugh, P., Preston, S.C. The Geometry of Axisymmetric Ideal Fluid Flows with Swirl. Arnold Math J. 3, 175–185 (2017). https://doi.org/10.1007/s40598-016-0058-2
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DOI: https://doi.org/10.1007/s40598-016-0058-2