Gromov's Convergence Theorem and Its Application
نویسندگان
چکیده
One of the basic questions of Riemannian geometry is that "If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?". Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), which may be useful not only for the above question but also beyond the above. But it seems to the author that his proof is heuristic and it contains some gaps (for these, see § 1), so we give a detailed proof of 8.25 in [7]. This is the first purpose of this paper. Second purpose is to prove a differentiable sphere theorem for manifolds of positive Ricci curvature, using the above theorem as a main tool. For a d-dimensional Riemannian manifold M, we denote by KM the sectional curvature, by vol (M) the volume, by diam (M) the diameter, by dM(m, n) the distance between m and n induced from Riemannian metric g and by iM the injectivity radius. A subset B is called d-dense when for any point me M, there exists a point n e B with dM{m, n) <ΞJ 3. A subset B is called ^-discrete if n19 n2e B (nx Φ n2) implies dM{nu n2) ^ δ. Let M(d, Δ, i0) (resp. M(d, Δ> p, v)) be the category of all complete Riemannian manifolds M with dimension = d, \KM\ <£ Δ and iM ^ i0 (resp. dimension = d, \KM\ <Ξ Δ, diam(M) ^ p, vol (M) ^ v). The following theorem is seemingly different from 8.25 in [7] but the inwardness is essentially same.
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