Conformal immersions of Riemannian products in low codimension
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Conformal immersions of Riemannian products in low codimension Felippe Guimarães1
· Bruno Mendonça2
Received: 13 November 2018 / Accepted: 6 March 2020 © Springer Nature B.V. 2020
Abstract We prove that conformal immersion of a Riemannian product M0n 0 × M1n 1 as a hypersurface in a Euclidean space must be an extrinsic product of immersions, under the assumption that n 0 , n 1 ≥ 2 and that M0n 0 × M1n 1 is not conformally flat. We also state a similar theorem for an arbitrary number of factors, more precisely, a conformal immersion f : M0n 0 × · · · × Mkn k → Rn+k must be an extrinsic product of immersions if one of the factors admits a plane with vanishing curvature and the remaining factors are not flat. Keywords Submanifolds · Riemannian products · Conformal immersions Mathematics Subject Classification 53B25
The simplest way to construct an immersion of a Riemannian product manifold into a Euclidean space is to take an extrinsic product of immersions, that is, a product of immersions of each factor of the product manifold. A natural problem in the submanifold theory is to provide sufficient conditions for an immersion of a Riemannian product to be decomposed as an extrinsic product, and there are plenty of works in this subject, for instance [1,2,5,7,8,10,12,14]. Some of them ask for intrinsic proprieties to decompose an arbitrary manifold, without the initial assumption that is already a product, in structures even more general than Riemannian products, as in [13,15]. In low codimension, it turns out that there is not enough space to the immersion of a Riemannian product manifold to be very complicated. Moore, in his outstanding work [10], showed that an immersion of a Riemannian product must be an extrinsic product when the
F. Guimarães: This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.
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Felippe Guimarães [email protected] Bruno Mendonça [email protected]
1
Universidade Federal de Sergipe, Av. Marechal Rondon s/n, Aracaju, SE 49100-000, Brazil
2
Universidade Estadual de Londrina, Rod Celso Garcia Cid PR 445 Km 380, Londrina, PR 86057-970, Brazil
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Geometriae Dedicata
factors are not flat and the codimension is the minimal possible. More precisely, he showed that Theorem 1 (Moore [10])Let f : M n → Rn+k be kan isometric immersion of a Riemannian k product manifold M n = i=0 Min i , where n = i=0 n i and n i ≥ 2 for all 1 ≤ i ≤ k. If the subset of points of Min i at which all sectional curvatures vanish has empty interior for all 1 ≤ i ≤ k, then M0n 0 is flat and f (M) is an open subset of a n 0 -cylinder over an extrinsic product of hypersurface immersions. The first step of his proof was to reduce the question to one of algebraic nature by proving an extrinsic decomposition De Rham type theorem. In the conformal realm, Tojeiro [14] proved an analogous theorem. In short, the aforementioned works of Tojeiro and Moore were the main motivation of this paper. We adapted the technique used by Moore to obtain the follow
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