Study of the structure of hypersurfaces with constant scalar curvature

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In the Euclidean spaces the known examples of complete hypersurfaces with positive constant scalar curvature are only spheres, generalized cylinders S^p×R^<n-p> and the rotational hypersurfaces. In this research we constructed infinitely many new examples of complete hypersurfaces with constant positive scalar curvature in the Euclidean spaces.For the O(p+1)×O(q+1)-invariant hypersurfaces, the equation representing its scalar curvature is constant S is(I){dx/ds=cosα, dy/ds=sinα, dα/ds=(p(p-1)((sinα)/x)^2-2pq(sinα)/x(cosα)/y+q(q-1)((cosα)/y)^2-S)/(2(q(cosα)/y-p(sinα)/x)),where α is the angle between the tangent vector (x', y') and the x-axis.The key point of this study is that we can compare the solution with another ODE which is explicitly integrable.Theorem Suppose that p 【less than or equal】 q + 1 and S > 0 (p 【greater than or equal】 2). Let 0 < x_0 【less than or equal】 √<p(p -1)/S> and 0 < y_0 【less than or equal】 √<q(q -1)/S>. Then the ODE system has a global solution γ(s) = (x(s), y(s)) ∈ R_+ × R_+ on (-∞, ∞) for the initial condition x(0) = x_0, y(0) = y_0 and α(0) = 0. Therefore M_γ become a complete hypersurface in R^<p+q+2> with constant scalar curvature S.Theorem Suppose that p > q + 1 and S > 0 (q 【greater than or equal】 2). Let 0 < x_0 【less than or equal】 √<(p -1)(q -1)/S> and 0 < y_0 【less than or equal】 √<q(q -1)/S>. Then the ODE system has a global solution γ(s) = (x(s), y(s)) ∈ R_+ × R_+ on (-∞, ∞) for the initial condition x(0) = x_0, y(0) = y_0 and α(0) = 0. Therefore M_γ become a complete hypersurface in R^<p+q+2> with constant scalar curvature S.