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Submanifolds of restricted type. (English) Zbl 0778.53002

A submanifold \(M^ n\) in a Euclidean space \(\mathbb{R}^ m\) is said to be of restricted type if the shape operator \(A_ H\), where \(H\) is the mean curvature vector, is the restriction of a fixed endomorphism \(A\) of \(\mathbb{R}^ m\) on the tangent space of \(M^ n\) at every point of \(M^ n\), i.e. \(A_ HX=(AX)^ T\), for any vector \(X\), tangent to \(M^ n\), where \((AX)^ T\) denotes the tangential component of \(AX\). There are many examples of submanifolds of restricted type. For example, every submanifold \(M^ n\) in \(\mathbb{R}^ m\) whose position vector field \(x\) satisfies \(\Delta x=Ax+B\), where \(\Delta\) is the Laplacian of \(M^ n\), \(A\) is a constant \((m\times m)\)-matrix and \(B\) a constant vector in \(\mathbb{R}^ m\), is of restricted type. It is interesting to ask which submanifolds are of restricted type? In this paper the authors prove that a hypersurface \(M^ n\) of a Euclidean space \(\mathbb{R}^{n+1}\) is of restricted type if and only if it is one of the following: a minimal hypersurface, an open part of a product hypersurface \(S^ k\times\mathbb{R}^{n-k}\) \((2\leq k\leq n)\), or an open part of a cylinder on a plane curve of restricted type. Moreover, the authors obtain a complete classification of plane curves of restricted type.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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