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We prove that all the spheres in the conformal 3-space have constant Gaussian curvature  $K=1$  if, and only if, the conformal factor is special. In this special case we study geometric properties of this ambient 3-space, and as an application we prove that it is isometric to the space ${\\mathbb{S}}^2\\times {\\mathbb{R}}$, so we consider it as the {\\em Radial Model} of ${\\mathbb{S}}^2\\times {\\mathbb{R}}$.  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Souza, Romildo Pina","submitted_at":"2016-06-27T20:31:06Z","abstract_excerpt":"In this work we study surfaces in radial conformally flat spaces. We characterize surfaces of rotation with constant Gaussian and Extrinsic curvature in these radial 3-spaces. We prove that all the spheres in the conformal 3-space have constant Gaussian curvature  $K=1$  if, and only if, the conformal factor is special. In this special case we study geometric properties of this ambient 3-space, and as an application we prove that it is isometric to the space ${\\mathbb{S}}^2\\times {\\mathbb{R}}$, so we consider it as the {\\em Radial Model} of ${\\mathbb{S}}^2\\times {\\mathbb{R}}$.  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