We discuss the necessary and su‰cient conditions for the existence of Lorentzian stationary surfaces in 4-dimensional space forms of index 2, and isometric stationary deformations preserving normal curvature.

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Abstract. Let XN{c) denote the TV-dimensional simply connected space form of constant curvature c. We consider a problem to classify those minimal surfaces in XN(c) which are locally isometric to minimal surfaces in X*{c). In this paper we solve this problem in the case where N = 4 , and give a result also in higher codimensional cases.

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Abstract.We improve the pinching theorem of Simons and the stability theorem of Barbosa and do Carmo with an elementary method.Simons [7] proved a pinching theorem for closed minimal submanifolds in the unit sphere, which led to an intrinsic rigidity result. In this note, using an elementary method, we improve his theorem and obtain a result which does not depend on the dimension of the ambient space.Lemma. Let M be an m-dimensional minimal submanifold in a space of constant curvature a. Let A and A denote the second fundamental form and the Laplacian of M, respectively. Then -(A,AA) <{2-2/(m -l)(m + 2)}\A\ -ma\A\2. Proof. We use the argument of Chern, do Carmo and Kobayashi [3]. We assume that the ambient space is «-dimensional. Set q = 2~lm(m + 1) -1 = 2~ (m -l)(m + 2). When n < m + q , the Lemma is included in [3]. So we assume that n > m + q in the following. We make a pointwise argument at a point p on M. Let {ex, ... ,en} be an orthonormal basis for the tangent space of the ambient space at p such that ex, ... ,em are tangent to M. We shall make use of the following convention on the ranges of indices: 1 < /., j < m , m+l < a, ß < n , m +1 < £, n < m + q. Let Ä" be the components of A with respect to the basis. Set T R = >~\ , h" Tzf. and T -T .It is an elementary observation that at each point the dimension of the image of the second fundamental form of an m-dimensional minimal submanifold is at most 2~ m(m+1 )-1 = q . Thus we may choose em+x, ... ,en so that ha-= 0 for a > m + q . Let F be a subspace of the normal space of M at p spanned by em+\. • • • >em+q ■ ^e define a symmetric linear transformation T of V by TCtZ^v^e) -Ys? n'P£nvr'ei' which lS weh defined. As T is symmetric, we may change em+x, ... ,em+q so that the (q x(?)-matrix (T() is diagonal. Then

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