Abstract
Some (not all) of the following generalizes to any dimension; this will be left to the reader. We will only give references for special topics. General references could be Berger and Gostiaux 1988 [175], Coxeter 1989 [409], do Carmo 1976 [451], Klingenberg 1995 [816], Spivak 1979 [1155], Sternberg 1983 [1157], and Stoker 1989 [1160]. For those who like computer programming, Gray 1998 [584] will be of interest. We will assume elementary calculus and also that functions are differentiable as often as needed.
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However, keep in mind the difficulty of making local measurements; angles between little pieces of straight lines are very sensitive to mismeasurement.
More precisely, rectifiable; see Wheeden & Zygmund 1977 [1258].
An interferometer is a device which uses the interference of two waves (radio, acoustic, or light waves will do) to make very precise distance measurements.
For the historian, it was discovered by Pinkall (see Karcher & Pinkall 1997 [782]) that the Whitney—Grauenstein theorem appears in Boy 1903 [249] as a footnote. We will meet Boy’s article again on page 136.
Peter Petersen points out that Schmidt’s proof in Schmidt 1939 [1105] uses Stokes’ theorem, even if obliquely.
Do not be afraid of elliptic functions. They are just the “classical functions” which come next after polynomials, rational fractions, exponentials and logarithms, trigonometric functions and their combinations.
Some people say closed instead of periodic, which can be ambiguous since it could also be used to mean only that a geodesic comes back to the same point, but not with the same direction; such a geodesic will not usually be periodic, and will be called a geodesic loop.
Real analytic functions are those equal to their convergent Taylor series, in an open set about each point. The definition of real analytic surfaces is analogous.
These surfaces are called developable for a reason to be seen below in §§1.6.7.
Also the opposite of concave, since concave versus convex is just a change of orientation, which doesn’t affect K.
Except, perhaps, this one.
The wave equation is just a first approximation. For better approximations, one needs to work much harder. For example, see Greenspan 1978 [598]. To our knowledge, no one has ever considered how to extend Greenspan’s work to curved surfaces, or to Riemannian geometry.
Recall that a stroboscope is an instrument which periodically flashes light, used for studying periodic motion.
A definition of Minkowski addition is presented in Berger 1994 [167] (11.1.3) or any book on convexity.
Also called the heat kernel.
Polyhedra are rather called polytopes starting with dimension d=4.
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© 2003 Springer-Verlag Berlin Heidelberg
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Berger, M. (2003). Old and New Euclidean Geometry and Analysis. In: A Panoramic View of Riemannian Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18245-7_1
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DOI: https://doi.org/10.1007/978-3-642-18245-7_1
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