Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton

By William Fulton

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5. 1 (Cheng [4]). If Mn is an oriented compact locally conformally flat hypersurface in E n + 1 , n > 3, with constant scalar curvature n(n — l)r, then Mn is isometric to a standard round sphere Sn(r). 5, we know that there are many complete hypersurfaces, which have two distinct principal curvatures, one of which is simple, but are not isometric to , S n _ 1 (c) x E and S 1 (c) x E " - 1 . 7 (Cheng [4]). Let Mn be an n-dimensional oriented complete hypersurface in E n + 1 with constant scalar curvature n(n — l)r and with two distinct principal curvatures one of which is simple.

So using diagonal arguments, we can glue these individual limiting connections on each good ball of the above covering of M \ EQO together, and we can obtain a subsequence {tm}, such that Am = ^rniAm) ^oo on whole M \ EQO, where the limiting connection A^, is defined on whole M\ETO. 4. Obviously, Aoo is a smooth Yang-Mills connection off E ^ . 1), JM as tm —• oo. Since the gauge-invariance of \JA„ | 2 , we have / \JAJ*< m lim f + JM \JAJ=0. , AQO is a Yang-Mills connection on M \ EQQ. D. §5. 1 in [CS]).

If Mn is an embedded compact hypersurface with constant scalar curvature in E™+1, then Mn is isometric to a standard round sphere. Proof. 5) since S = Yli=i tf. ^ nH2 holds, where A, are the principal curvatures of M " . Since M " is a compact hypersurface in E n + 1 . Let denote the position vector field of E n + 1 relative to the origin. 6) D(4>, ) = n(n - 1)H + n(n - l)r(cj>, N). 7) / (n(n-l)H + n(n-l)r((f>,N))dMn = 0, where dMn denotes the volume element on Mn. Since Mn is a compact embedded hypersurface in E n + 1 .

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