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A characterization of harmonic sections and a Liouville theorem

Simão Stelmastchuk (2012)

Archivum Mathematicum

Let P ( M , G ) be a principal fiber bundle and E ( M , N , G , P ) an associated fiber bundle. Our interest is to study the harmonic sections of the projection π E of E into M . Our first purpose is give a characterization of harmonic sections of M into E regarding its equivariant lift. The second purpose is to show a version of a Liouville theorem for harmonic sections of π E .

A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken (2017)

Czechoslovak Mathematical Journal

If ( M , ) is a manifold with a symmetric linear connection, then T * M can be endowed with the natural Riemann extension g ¯ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to g ¯ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure 𝒫 on ( T * M , g ¯ ) and prove that 𝒫 is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if g ¯ reduces to the...

A short note on f -biharmonic hypersurfaces

Selcen Y. Perktaş, Bilal E. Acet, Adara M. Blaga (2020)

Commentationes Mathematicae Universitatis Carolinae

In the present paper we give some properties of f -biharmonic hypersurfaces in real space forms. By using the f -biharmonic equation for a hypersurface of a Riemannian manifold, we characterize the f -biharmonicity of constant mean curvature and totally umbilical hypersurfaces in a Riemannian manifold and, in particular, in a real space form. As an example, we consider f -biharmonic vertical cylinders in S 2 × .

Almost symplectic structures and harmonic morphisms

Jean-Marie Burel (2004)

Bollettino dell'Unione Matematica Italiana

In this paper, we introduce the notion of symplectic harmonic maps between tamed manifolds and establish some properties. In the case where the manifolds are almost Hermitian manifolds, we obtain a new method to contruct harmonic maps with minimal fibres. We finally present examples of such applications between projectives spaces.

Anti-invariant Riemannian submersions from almost Hermitian manifolds

Bayram Ṣahin (2010)

Open Mathematics

We introduce anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and check the harmonicity of such submersions. We also find necessary and sufficient conditions for a Langrangian Riemannian submersion, a special anti-invariant Riemannian submersion, to be totally geodesic. Moreover, we obtain decomposition theorems for the total manifold...

Asymptotic values of minimal graphs in a disc

Pascal Collin, Harold Rosenberg (2010)

Annales de l’institut Fourier

We consider solutions of the prescribed mean curvature equation in the open unit disc of euclidean n-dimensional space. We prove that such a solution has radial limits almost everywhere; which may be infinite. We give an example of a solution to the minimal surface equation that has finite radial limits on a set of measure zero, in dimension two. This answers a question of Nitsche.

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