Conformal Riemannian Maps between Riemannian Manifolds, Their Harmonicity and Decomposition Theorems
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Conformal Riemannian Maps between Riemannian Manifolds, Their Harmonicity and Decomposition Theorems Bayram S.ahin
Received: 23 June 2008 / Accepted: 22 October 2008 / Published online: 31 October 2008 © Springer Science+Business Media B.V. 2008
Abstract Riemannian maps were introduced by Fischer (Contemp. Math. 132:331–366, 1992) as a generalization isometric immersions and Riemannian submersions. He showed that such maps could be used to solve the generalized eikonal equation and to build a quantum model. On the other hand, horizontally conformal maps were defined by Fuglede (Ann. Inst. Fourier (Grenoble) 28:107–144, 1978) and Ishihara (J. Math. Kyoto Univ. 19:215– 229, 1979) and these maps are useful for characterization of harmonic morphisms. Horizontally conformal maps (conformal maps) have their applications in medical imaging (brain imaging)and computer graphics. In this paper, as a generalization of Riemannian maps and horizontally conformal submersions, we introduce conformal Riemannian maps, present examples and characterizations. We show that an application of conformal Riemannian maps can be made in weakening the horizontal conformal version of Hermann’s theorem obtained by Okrut (Math. Notes 66(1):94–104, 1999). We also give a geometric characterization of harmonic conformal Riemannian maps and obtain decomposition theorems by using the existence of conformal Riemannian maps. Keywords Isometric immersion · Riemannian submersion · Horizontally conformal submersion · Riemannian map · Conformal Riemannian map Mathematics Subject Classification (2000) 53C20 · 53C43 · 53C12
1 Introduction The theory of smooth maps between Riemannian manifolds has been widely studied in Riemannian geometry. Such maps are useful for comparing geometric structures between two manifolds. In this point of view, isometric immersions are basic such maps between Riemannian manifolds and they are characterized by their Riemannian metrics and Jacobian matrices. More precisely, a smooth map F : (M1m , g1 ) −→ (M2n , g2 ) between Riemannian B. S.ahin () Department of Mathematics, Inonu University, 44280, Malatya, Turkey e-mail: [email protected]
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manifolds (M1m , g1 ) and (M2n , g2 ), where dim(M1 ) = m and dim(M2 ) = n, is called an isometric immersion if F∗ is injective and g2 (F∗ X, F∗ Y ) = g1 (X, Y )
(1.1)
for X, Y vector fields tangent to M1m , here F∗ denotes the derivative map. It is known that the theory of isometric immersions originated from Gauss’s studies on surfaces of Euclidean spaces. On the other hand, the study of Riemannian submersions between Riemannian manifolds was initiated by O’Neill [31] and Gray [18], see also [12] and [41]. A smooth map F : (M1m , g1 ) −→ (M2n , g2 ) is called an Riemannian submersion if F∗ is onto and it satisfies (1.1) for vector fields tangent to the horizontal space (ker F∗ )⊥ . The simplest example of a Riemannian submersion is the projection of a Riemannian product manifold on one of its factors. In 1992, Fischer introduced Riemannian maps between Riemannian manif
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