Let Mn (n ⩾ 3) be a complete Riemannian manifold with secM ⩾ 1, and let $$M_i^{n_i }$$ M i n i (i = 1, 2) be two complete totally geodesic submanifolds in M. We prove that if n1 + n2 = n − 2 and if the distance |M1M2| ⩾ π/2, then Mi is isometric to $$\mathbb{S}^{n_i } /\mathbb{Z}_h$$ S n i / Z h , $$\mathbb{C}P^{n_i /2}$$ C P n i / 2 , or $$\mathbb{C}P^{n_i /2} /\mathbb{Z}_2 $$ C P n i / 2 / Z 2 with the canonical metric when ni > 0; and thus, M is isometric to Sn/ℤh, ℂPn/2, or ℂPn/2/ℤ2 except possibly when n = 3 and $$M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{S}^1 /\mathbb{Z}_h $$ M 1 ( o r M 2 ) ≅ i s o S 1 / Z h with h ⩾ 2 or n = 4 and $$M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{R}P^2 $$ M 1 ( o r M 2 ) ≅ i s o R P 2 .