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I would like to ask the following question: Suppose an $m$-dimensional submanifold in $\mathbf{R}^n$, such that there is a constant $l$, representing the largest number allowing an open normal bundle around this manifold of radius $r<l$ to be embedded in $\mathbf{R}^n$. Now if i establish some fixed sized grid in $\mathbf{R}^n$ and then pick one the cells that contains some part of the manifold. Suppose the volume of the manifold in this cell is $V_0$, what can i tell about the bounds for the different types of curvatures (scalar, mean, sectional curvatures) for the manifold inside this ball? Thanks very much in advance!

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In particular the second fundamental form of you submanifold is bounded; therefore sectional curvature is bounded (no need to pass to a grid). –  Anton Petrunin Aug 20 '13 at 17:49
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