Warped Product Of S ~ 1 (a) ¡Á _fs ~ N (b) Some Results

This paper includes three chapters.In Chapter one, we define a family {g_t}_t∈R⁺ of Riemannian metrics on I× S³, where I is the interval (0,l/) or (0, +∞). By investigating these Riemannian metrics's properties about the Schouten functional on I × S³, we obtain the critical metric's result of the Schouten functional on S¹ × S³.In Chapter two, we study warped product S¹(a) × _f Sⁿ(b) (a² + b² = 1, a > 0,b > 0, n≥ 3), and obtain a necessary and sufficient condition of this kind of Riemannian manifolds which have positive isotropic curvature.In Chapter three, in M. J. Gursky's paper, we point out some problems and modify a lemma and a corollary. Thus some remarks are obtained.Now, we can state our main results as follows.Theorem A (see (1.1.4)and(1.1.5)) For (?)t > 0, g_t is not the non-trivial critical metric of the Schouten functional on I × S³.Theorem B (see (1.1.4)and(1.1.5)) On S¹ × S³, there is no non-trivial critical metric of the Schouten functional which likes g_t (Where f(r) is a periodic function.).Where we call a critical metric non-trivial if it is neither locally conformally flat nor Einstein.Theorem C Warped product (S¹(a) × _fSⁿ(b), g|⁾ have positive isotropic curvature if and only if f satisfies ('1/2f²)″ < (1/(b²)) and -1/b < f′ < 1/b. Where g = ds₁² + f²ds_n². ds₁² and ds_n² denote the canonical metric on S¹(a) (?) R² and on Sⁿ(b) (?) Rⁿ⁺¹, respectively.