Construction of Harmonic Maps between Semi-Riemannian Spheres (original) (raw)

Helical geodesic immersions of semi-Riemannian manifolds

Kodai Mathematical Journal, 2007

We obtain some basic results on helical geodesic immersions in semi-Riemannian geometry. For example, it is shown that, for an indefinite semi-Riemannian submanifold, if any space-like geodesics of the submanifold are helices of order d, curvatures l 1 ;. .. ; l dÀ1 and signatures e 1 ;. .. ; e d in the ambient space, then any timelike geodesics of the submanifold have the same order and curvatures, and signatures ðÀ1Þ 1 e 1 ;. .. ; ðÀ1Þ d e d .

Exponentially Harmonic Maps into Spheres

Axioms

We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere S m ⊂ R m + 1 . Given a codimension two totally geodesic submanifold Σ ⊂ S m , we show that every nonconstant exponentially harmonic map ϕ : M → S m either meets or links Σ . If H 1 ( M , Z ) = 0 then ϕ ( M ) ∩ Σ ≠ ∅ .

Harmonic morphisms between riemannian manifolds

Annales de l’institut Fourier, 1978

A smooth map f : M → N between semi-riemannian manifolds is called a harmonic morphism if f pulls back harmonic functions (i.e., local solutions of the Laplace-Beltrami equation) on N into harmonic functions on M. It is shown that a harmonic morphism is the same as a harmonic map which is moreover horizontally weakly conformal, these two notions being likewise carried over from the riemannian case. Additional characterizations of harmonic morphisms are given. The case where M and N have the same dimension n is studied in detail. When n = 2 and the metrics on M and N are indefinite, the harmonic morphisms are characterized essentially by preserving characteristics.

Isometric Immersions with Geodesic Normal Sections in Semi-Riemannian Geometry

Tokyo Journal of Mathematics, 2008

We study an isometric immersion f : M →M with geodesic normal sections, whereM is a semi-Riemannian space form. In Riemannian geometry, it is known that f is helical, in particular, every geodesics of M have the same proper order inM. However this does not hold in general, whenM is indefinite semi-Riemannian. We give sufficient conditions for an isometric immersion with geodesic normal sections to be helical.

Harmonic Gauss maps

Pacific Journal of Mathematics, 1989

A construction is given whereby a Riemannian manifold induces a Riemannian metric on the total space of a large class of fibre bundles over it. Using this metric on the appropriate bundles, necessary and sufficient conditions are given for the Gauss map and the spherical Gauss map to be harmonic. A weak maximum principle is applied to the Gauss map of an isometric immersion into Euclidean space in order to prove a sufficient condition for when such an immersion with parallel mean curvature vector must be minimal.

Harmonic maps and isometric embeddings of the spacetime

Physics Letters A, 2004

We investigate harmonic maps in the context of isometric embeddings when the target space is Ricci-flat and has codimension one. With the help of the Campbell-Magaard theorem we show that any n-dimensional (n 3)

P. Baird, J. C. Wood : Harmonic Morphisms between Riemannian Manifolds (London Mathematical Society Monographs: New Series 29)

Book Review

book reviews 869 spherical and hyperbolic space. It is divided into two equal-sized parts: the first is devoted to the two-dimensional case, where much more is known than in the n-dimensional setting, which is discussed in the second part. In addition, there is an appendix providing some important background information, essentially from convex geometry. Many of the sections end with interesting and stimulating open problems, and each chapter closes with a brief survey of related problems.

1 Local Symmetry of Harmonic Spaces as Determined by the Spectra of Small Geodesic Spheres

2016

We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm |∇R| of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric. For the proof we use the first few heat invariants and consider certain coefficients in the radial power series expansions of the curvature invariants |R| 2 and |Ric| 2 of the geodesic spheres. Moreover, we obtain analogous results for geodesic balls with either Dirichlet or Neumann boundary conditions. We also comment on the relevance of these results to constructions of Z.I. Szabó.