By Marcel Berger

This ebook introduces readers to the dwelling themes of Riemannian Geometry and information the most effects recognized to this point. the implications are said with no specific proofs however the major principles concerned are defined, affording the reader a sweeping panoramic view of virtually the whole lot of the sphere.

From the stories ''The e-book has intrinsic price for a scholar in addition to for an skilled geometer. also, it truly is a compendium in Riemannian Geometry.'' --MATHEMATICAL REVIEWS

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**Example text**

L p L q these two curves are isometric Fig. 11. These two curves are isometric An important remark is in order: the attentive reader might have been puzzled, even outraged, that we never distinguished between a curve as a map t → c(t) from an interval of the real line, and its image as the set of points c(t) when t ranges through the values of that interval. 3 Plane Curves 11 are obscure even in many books. We refer the interested reader to chapter 8 of Berger and Gostiaux 1988 [175] for a very detailed exposition of this point.

Let us look at this again, now in a more sophisticated way. We start with a curve c with ends p = c(a) and q = c(b). Suppose it is as short as any curve can be with these extremities. Consider any one parameter family of curves cμ with the same ends, with c = c0 , and compute their lengths. The fact that the length should be a minimum implies in particular that the ﬁrst derivative of the length as a function of the parameter μ has to vanish. Computation yields 1 Euclidean Geometry 16 Fig. 17. 3) a μ=0 where the vector valued function f (s) (which we can choose to be orthogonal to the curve everywhere) estimates the derivative of the normal displacement of the family of curves.

To rest a little, we mention an interesting fact rarely found in textbooks. When at a point m of the curve c, the curvature varies, that is to 14 1 Euclidean Geometry Fig. 15. Finding tangent line, concavity and curvature say dK/dt(m) = 0, then the osculating circle at m crosses the curve (guess where it is inside, and where outside). A baﬄing consequence is that the osculating circles of of a curve whose curvature is never critical never intersect one another. 15. This picture gives two counterexamples.