Totally Real Parallel Siibmanifolcls In CP₁ⁿ（c）

In the present article, we mainly study the totally real parallel submanifolds inpseudo-riemannian symmetric space CPn1（c）. We investigate three problems, and theseproblems are respectively discussed in section2, section3, section4and5.In section1, firstly we introduce the background of submanifolds and symmetricspace, then we give some fundamental theory about pseudo-riemannian manifold andtotally real parallel submanifolds of CP₁ⁿ（c）.In section2, we introduce the setMof symmetric trilinear forms satisfying certainconditions. Then we define an equivariant immersion fs,σ associated to trilinear form σ.The associated G-equivariant immersion fs,σ is a totally real parallel isometric immersionof Mn1into CP₁ⁿ（c）.In section3, we denote by（?）Mthe set of all equivalence classes of totally real parallelisometric immersions of M₁ⁿinto CP₁ⁿ（c）,（?）Mthe set of all equivalence classes of com-plete totally real parallel submanifolds in CP₁ⁿ（c） with the universal pseudo-riemanniancovering M₁ⁿand （?）Mthe set of all equivalence classes of the trilinear forms. We showthat there are the natural correspondences among these sets（?）M,（?）M, and （?）M.In section4and5, we study the setMfor a pseudo-riemannian symmetric spaceM_rⁿand an interesting example in the geometry of totally real surfaces in CP₁²（c）.