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In this paper, we give a definition of coherent tangent bundles of space form type, which is a generalized notion of space forms. Then, we classify their realizations in the sphere as a wave front, which is a generalization of a theorem of O’Neill and Stiel: any isometric immersion of the n-sphere into the ((n+1))-sphere of the same sectional curvature is totally geodesic.

Isometric Immersions with Singularities Between Space Forms of the Same Positive Curvature