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Studyng geometry of some Riemannian manifolds with a help of a computer


Specific goal of the project is to use symbolic computation software developed for personal computers in studying geometrical properties of some riemannian manifolds, which are defined explicitly in coordinate forms. For this purpose we suppose to use the programs "Maple Y", "Derive" and "Reduce" for 80486 - Based Systems PC/MS DOS with a mathematical co-processor. First, we suppose to consider total spaces Pn and Pn of principal S3 bundles over S4 and S7. According to [A3] we suppose to give their representations in coordinates, compute their first and second fundamental forms, and, if possible, to estimate a sign of their curvature. In case when the curvature is nonnegative, this will give new examples of nonnegatively curved manifolds and also an affirmative answer to a well know conjecture that all vector bundles over a standard sphere admit metrics of nonnegative sectional curvature (of course, for the considered case: bundles over S4 and S7). The second purpose of the project is to compute Ricci curvature of some warp-products of spheres and euclidean spaces, which give examples of positively Ricci curved open manifolds. This may help to verify the following conjectures on uniqueness of a tangent cone at infinity: 1. Do all ideal boundaries of a given open manifold of nonnegative Ricci curvature have same dimension? 2. Does a given open manifold of nonnegative Ricci curvature have unique ideal boundary if its curvature decay is bigger then the inverse square? (AU)

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