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Marcos Salvai 《Annals of Global Analysis and Geometry》1998,16(4):357-370
Recent results of Gordon, Mao and Pesce imply that isospectral compact hyperbolic Riemann surfaces have Laplace and length isospectral unit tangent bundles. In this note we give explicit formulae relating the spectra of such surfaces and those of their unit tangent bundles, and use them to prove the converses. 相似文献
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Marcos Salvai 《Monatshefte für Mathematik》2014,175(4):613-619
Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic. 相似文献
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We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold equipped with the Sasaki metric and discuss the characterising condition for critical points. Furthermore, we provide a useful
method for computing the tension field in some particular situations. Such a method is shown to be adequate for many tensor
fields defined on manifolds M equipped with a G-structure compatible with . This leads to the construction of several new examples of differential forms which are harmonic sections or determine a
harmonic map from into its sphere bundle. 相似文献
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Marcos Salvai 《Monatshefte für Mathematik》2006,147(2):173-177
The Fréchet manifold
of all embeddings (up to orientation preserving reparametrizations) of the circle in S
3 has a canonical weak Riemannian metric. We use the characterization obtained by H. Gluck and F. Warner of the oriented great
circle fibrations of S
3 to prove that among all such fibrations π:S
3→B, the manifold B consisting of the oriented fibers is totally geodesic in
, or has minimum volume or diameter with the induced metric, exactly when π is a Hopf fibration.
Partially supported by foncyt, Antorchas, ciem (conicet) and secyt (unc). 相似文献
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Let M be a compact Riemannian manifold and let μ, d be the associated measure and distance on M. Robert McCann, generalizing results for the Euclidean case by Yann Brenier, obtained the polar factorization of Borel maps S: M → M pushing forward μ to a measure ν: each S factors uniquely a.e. into the composition S = T ? U, where U: M → M is volume preserving and T: M → M is the optimal map transporting μ to ν with respect to the cost function d2/2.In this article we study the polar factorization of conformal and projective maps of the sphere S n . For conformal maps, which may be identified with elements of Oo(1, n+1), we prove that the polar factorization in the sense of optimal mass transport coincides with the algebraic polar factorization (Cartan decomposition) of this Lie group. For the projective case, where the group GL+(n + 1) is involved, we find necessary and sufficient conditions for these two factorizations to agree. 相似文献
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Marcos Salvai 《Monatshefte für Mathematik》2006,117(1):173-177
The Fréchet manifold
E/ ~ {\cal E}/_{\!\sim}
of all embeddings (up to orientation preserving reparametrizations) of the circle in S
3 has a canonical weak Riemannian metric. We use the characterization obtained by H. Gluck and F. Warner of the oriented great
circle fibrations of S
3 to prove that among all such fibrations π:S
3→B, the manifold B consisting of the oriented fibers is totally geodesic in
E/ ~ {\cal E}/_{\sim }
, or has minimum volume or diameter with the induced metric, exactly when π is a Hopf fibration. 相似文献
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Suppose that the sphere Sn has initially a homogeneous distribution of mass and let G be the Lie group of orientation preserving projective diffeomorphisms of Sn. A projective motion of the sphere, that is, a smooth curve in G, is called force free if it is a critical point of the kinetic energy functional. We find explicit examples of force free projective motions of Sn and, more generally, examples of subgroups H of G such that a force free motion initially tangent to H remains in H for all time (in contrast with the previously studied case for conformal motions, this property does not hold for H=SOn+1). The main tool is a Riemannian metric on G, which turns out to be not complete (in particular not invariant, as happens with non-rigid motions), given by the kinetic energy. 相似文献