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On the Finsler geometry of the Heisenberg group H 2 n + 1 and its extension

Mehri Nasehi (2021)

Archivum Mathematicum

We first classify left invariant Douglas ( α , β ) -metrics on the Heisenberg group H 2 n + 1 of dimension 2 n + 1 and its extension i.e., oscillator group. Then we explicitly give the flag curvature formulas and geodesic vectors for these spaces, when equipped with these metrics. We also explicitly obtain S -curvature formulas of left invariant Randers metrics of Douglas type on these spaces and obtain a comparison on geometry of these spaces, when equipped with left invariant Douglas ( α , β ) -metrics. More exactly, we show...

On the first eigenvalue of spacelike hypersurfaces in Lorentzian space

Bing Ye Wu (2006)

Archivum Mathematicum

In this paper we obtain a lower bound for the first Dirichlet eigenvalue of complete spacelike hypersurfaces in Lorentzian space in terms of mean curvature and the square length of the second fundamental form. This estimate is sharp for totally umbilical hyperbolic spaces in Lorentzian space. We also get a sufficient condition for spacelike hypersurface to have zero first eigenvalue.

On the first secondary invariant of Molino's central sheaf

Jesús A. Álvarez López (1996)

Annales Polonici Mathematici

For a Riemannian foliation on a closed manifold, the first secondary invariant of Molino's central sheaf is an obstruction to tautness. Another obstruction is the class defined by the basic component of the mean curvature with respect to some metric. Both obstructions are proved to be the same up to a constant, and other geometric properties are also proved to be equivalent to tautness.

On the four vertex theorem in planes with radial density e φ ( r )

Doan The Hieu, Tran Le Nam (2008)

Colloquium Mathematicae

It is shown that in a plane with a radial density the four vertex theorem holds for the class of all simple closed curves if and only if the density is constant. On the other hand, for the class of simple closed curves that are invariant under a rotation about the origin, the four vertex theorem holds for every radial density.

On the Gauss map of B-scrolls in 3 -dimensional Lorentzian space forms

Angel Ferrández, Pascual Lucas (2000)

Czechoslovak Mathematical Journal

In this note we show that B -scrolls over null curves in a 3-dimensional Lorentzian space form M ¯ 1 3 ( c ) are characterized as the only ruled surfaces with null rulings whose Gauss maps G satisfy the condition Δ G = Λ G , Λ X ( M ¯ ) → X ( M ¯ ) being a parallel endomorphism of X ( M ¯ ) .

On the generic spectrum of a riemannian cover

Steven Zelditch (1990)

Annales de l'institut Fourier

Let M be a compact manifold let G be a finite group acting freely on M , and let ℳ G be the (Fréchet) space of G -invariant metric on M . A natural conjecture is that, for a generic metric in ℳ G , all eigenspaces of the Laplacian are irreducible (as orthogonal representations of G ). In physics terminology, no “accidental degeneracies” occur generically. We will prove this conjecture when dim M ≥ dim V for all irreducibles V of G . As an application, we construct isospectral manifolds with simple eigenvalue...

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