Let M (n) (n a (c) 3/4 3) be a complete Riemannian manifold with sec (M) a (c) 3/4 1, and let (i =3D 1, 2) be two complete totally geodesic submanifolds in M. We prove that if n (1) + n (2) =3D n - 2 and if the distance |M (1) M (2)| a (c) 3/4 pi/2, then M (i) is isometric to , , or with the canonical metric when n (i) > 0; and thus, M is isometric to S (n) /a (h) , a",P (n/2), or a",P (n/2)/a(2) except possibly when n =3D 3 and with h a (c) 3/4 2 or n =3D 4 and . Rp(2)
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