On Isometric and Conformal Rigidity of Submanifolds

Let f, g : Mn → Rn+d be two immersions of an n-dimensional differentiable manifold into Euclidean space. That g is conformal (isometric) to f means that the metrics induced on Mn by f and g are conformal (isometric). We say that f is conformally (isometrically) rigid if given any other conformal (isometric) immersion g there exists a conformal (isometric) diffeomorphism Υ from an open subset of Rn+d to an open subset of Rn+d such that g = Υ ◦ f. In this case, we say that f and g are conformally (isometrically) congruent. It is then an interesting problem to determine conditions on f which imply conformal (isometric) rigidity. E. Cartan ([Ca1] , see also [Da]) showed that when n ≥ 5 a hypersurface f : Mn → Rn+1 is “generically” conformally rigid. To be more specific, he proved that f is conformally rigid when the maximal dimension of an umbilical subspace is at most n − 3 at any point. Later, do Carmo and Dajczer ([C-D]) introduced a conformal invariant for immersions of arbitrary codimension, namely, the conformal s-nullity νc s , and generalized Cartan’s result. More precisely, they showed that conformal rigidity holds whenever d ≤ 4 , n ≥ 2d + 3 and νc s ≤ n − 2s − 1 for 1 ≤ s ≤ d. As far as we know, it is still an open problem whether this result remains true for any codimension d. In this paper, we introduce a new conformal invariant, namely, the conformal type number τ c f , and prove the following result which has no restriction on the size of the codimension.