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By Luther Pfahler Eisenhart

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However, the local factor, C, need not be global. To see this, consider the case of a 2-dimensional cylinder R x 8 1 = C 2 /Z. If we view the real line, R, as the collapsed Gromov-Hausdorff limit of a sequence of cylinders for which the 8 1 factor shrinks to a point, we see that the local isometric factor corresponding to J(v) (the 8 1 factor) can disappear in the limit. 4) which are metric cones, the above complications do not occur. Roughly speaking, an almost parallel vector field on a cone is always almost the gradient of the coordinate function corresponding to the factor, R, of some (global) almost isometric splitting.

Intuitively, tangent cones are obtained by blowing up the space at a point and taking a limit, possibly after passing to a subsequence. In case Y happens to be a smooth riemannian manifold, this proceedure recovers the usual tangent space with its canonical metric. JEFF CHEEGER 48 Let P denote the distance function (Le. the metric) of Y. Gromov's compactness theorem implies that every pointed sequence, {(Y, y, ri l pH, has a subsequence which converges in the pointed Gromov-Hausdorff sense, to some space (Yy, Yoo, Poo).

2) where Q lu . ,I. 8g", ,I' one Bz Zn'& 8z m, ' of k,l,m, equals i, and one of k',l',m' equals j. 2), in harmonic coordinates, the Einstein condition becomes a quasi-linear elliptic equation on the metric. TR • , Relative Volullle cOlllparison. Let M'H denote the complete, simply connected, n-dimensional space of constant curvature == H. Let l!. 3. ([Bish], [GvLPD If RicM" functions are nondecreasing: Vol(8Br(P)) -I. t. Monotonicity inequalities playa key role in geometric analysis. They are assertions to the effect that behavior of a certain quantity on a fixed scale, controls its behavior on all smaller scales.

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