ISSN:
1420-8946
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. Let M n be a compact (two-sided) minimal hypersurface in a Riemannian manifold $ \bar M^{n+1} $ . It is a simple fact that if $ \bar M $ has positive Ricci curvature then M cannot be stable (i.e. its Jacobi operator L has index at least one). If $ \bar M = S^{n+1} $ is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator.¶We prove that if $ \bar M $ is the real projective space $ P^{n+1} = S^{n+1}/\{\pm\} $ , obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface $ S^{n_1}(R_1) \times S^{n_2}(R_2) \subset S^{n+1} $ obtained as the product of two spheres of dimensions n 1, n 2 and radius R 1, R 2, with $ n_1 + n_2 = n, R_1^2 + R_2^2 = 1 $ and $ n_1R_2^2 = n_2R_1^2 $ .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/PL00000373
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