“The treatises of Darboux (1842–1917) and Bianchi (1856–1928) on surface theory are among the great works in the mathematical literature. They are: G. Darboux, Théorie générale des surfaces, Tome 1 (1887), 2 (1888), 3 (1894), 4 (1896), and later editions and reprints. L. Bianchi. Lezioni di Geometria Differenziale, Pisa 1894; German translation by Lukat, Lehrbuch der Differentialgeometrie, 1899. The subject is basically local surface theory.”

—  Shiing-Shen Chern

[1991, Surface Theory with Darboux and Bianchi, Miscellanea Mathematica, 59–69, Springer, https://doi.org/10.1007/978-3-642-76709-8_4]

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "The treatises of Darboux (1842–1917) and Bianchi (1856–1928) on surface theory are among the great works in the mathema…" by Shiing-Shen Chern?
Shiing-Shen Chern photo
Shiing-Shen Chern 6
mathematician (1911–2004), born in China and later acquirin… 1911–2004

Related quotes

“All the light which is radiated… will, after it has traveled a distance r, lie on the surface of a sphere whose area S is given by the first of the formulae (3). And since the practical procedure… in determining d is equivalent to assuming that all this light lies on the surface of a Euclidean sphere of radius d, it follows…4 \pi d^2 = S = 4 \pi r^2 (1 - \frac{K r^2}{3} + …);whence, to our approximation 4)d = r (1- \frac{K r^2}{6} + …), or
r = d (1 + \frac{K d^2}{6} + …).</center”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces… The intrinsic geometry of such a space of curvature K provides formulae for the surface area S and the volume V of a "small sphere" of radius r, whose leading terms are 3)S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + …),
V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + …).”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“Measurements which may be made on the surface of the earth… is an example of a 2-dimensional congruence space of positive curvature K = \frac{1}{R^2}… [C]onsider… a "small circle" of radius r (measured on the surface!)… its perimeter L and area A… are clearly less than the corresponding measures 2\pi r and \pi r^2… in the Euclidean plane. …for sufficiently small r (i. e., small compared with R) these quantities on the sphere are given by 1):L = 2 \pi r (1 - \frac{Kr^2}{6} + …),
A = \pi r^2”

Howard P. Robertson (1903–1961) American mathematician and physicist

1 - \frac{Kr^2}{12} + …
Geometry as a Branch of Physics (1949)

Robbert Dijkgraaf photo

“The last two years have seen the emergence of a beautiful new subject in mathematical physics. It manages to combine a most exotic range of disciplines: two-dimensional quantum field theory, intersection theory on the moduli space of Riemann surfaces, integrable hierarchies, matrix integrals, random surfaces, and many more. The common denominator of all these fields is two-dimensional quantum gravity or, more general, low-dimensional string theory.”

Robbert Dijkgraaf (1960) Dutch mathematical physicist and string theorist

[1992, Intersection Theory, Integrable Hierarchies and Topological Field Theory by Robbert Dijkgraaf, Fröhlich J., ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds.), New Symmetry Principles in Quantum Field Theory, NATO ASI Series (Series B: Physics), vol. 295, 95–158, Springer, Boston, MA, 10.1007/978-1-4615-3472-3_4]

Alfred Marshall photo

“I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypothesis was very well unlikely to be good economics: and I went more and more on the rules - (1) Use mathematics as shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life (5) Burn the mathematics. (6) If you can’t succeed in 4, burn 3. This last I do often.”

Alfred Marshall (1842–1924) British economist

Letter to A.L. Bowley, 27 February 1906, cited in: David L. Sills, Robert King Merton, Social Science Quotations: Who Said What, When, and Where http://books.google.com/books?id=WIKQbew5YKcC&pg=PA151 Transaction Publishers, 2000. p. 151.

“A discipline has six basic characteristics:
(1) a focus of study,
(2) a world view or paradigm,
(3) a set of reference disciplines used to establish the discipline,
(4) principles and practices associated with the discipline,
(5) an active research or theory development agenda, and
(6) the deployment of education and promotion of professionalism”

Donald H. Liles (1947) American engineer

Source: The Enterprise Engineering Discipline (1996), p. 1

Edward Witten photo

“Replacing particles by strings is a naive-sounding step, from which many other things follow. In fact, replacing Feynman graphs by Riemann surfaces has numerous consequences: 1. It eliminates the infinities from the theory. …2. It greatly reduces the number of possible theories. …3. It gives the first hint that string theory will change our notions of spacetime. Just as in QCD, so also in gravity, many of the interesting questions cannot be answered in perturbation theory. In string theory, to understand the nature of the Big Bang, or the quantum fate of a black hole, or the nature of the vacuum state that determines the properties of the elementary particles, requires information beyond perturbation theory… Perturbation theory is not everything. It is just the way the [string] theory was discovered.”

Edward Witten (1951) American theoretical physicist

"The Past and Future of String Theory" in The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking's Contributions to Physics (2003) ed. G.W. Gibbons, E.P.S. Shellard & S.J. Rankin

“There [is]… a list of five tests which any body of doctrine which aspires to the name of economic theory must be able to meet:
1. Economic theory should unify economic science. - The task of a general body of theory in any subject is to give unity to its investigations…
2. Economic theory should be relevant to the modern problem of control. - Students of economics should spend their efforts upon subjects worth investigation…
3. The proper subject-matter of economic theory is institutions. - The demand that economic theory relate to institutions is implicit in the plea for its relevancy…
4. Economic theory is concerned with matters of process. - If economic theory is to treat of institutions it must know both the kinds of things institutions are and the kinds of things they are not…
5. Economic theory must be based upon an acceptable theory of human behavior.”

Walton Hale Hamilton (1881–1958) Yale Law Professor

After all control and institutions and processes are immediate things. They can all be translated into terms of human conduct...
Source: The Institutional Approach to Economic Theory, 1919, p. 311-6

Hans Reichenbach photo

“The surfaces of three-dimensional space are distinguished from each other not only by their curvature but also by certain more general properties. A spherical surface, for instance, differs from a plane not only by its roundness but also by its finiteness. Finiteness is a holistic property. The sphere as a whole has a character different from that of a plane. A spherical surface made from rubber, such as a balloon, can be twisted so that its geometry changes…. but it cannot be distorted in such a way as that it will cover a plane. All surfaces obtained by distortion of the rubber sphere possess the same holistic properties; they are closed and finite. The plane as a whole has the property of being open; its straight lines are not closed. This feature is mathematically expressed as follows. Every surface can be mapped upon another one by the coordination of each point of one surface to a point of the other surface, as illustrated by the projection of a shadow picture by light rays. For surfaces with the same holistic properties it is possible to carry through this transformation uniquely and continuously in all points. Uniquely means: one and only one point of one surface corresponds to a given point of the other surface, and vice versa. Continuously means: neighborhood relations in infinitesimal domains are preserved; no tearing of the surface or shifting of relative positions of points occur at any place. For surfaces with different holistic properties, such a transformation can be carried through locally, but there is no single transformation for the whole surface.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“We see that each surface is really a pair of surfaces, so that, where they appear to merge, there are really four surfaces. Continuing this process for another circuit, we see that there are really eight surfaces etc and we finally conclude that there is an infinite complex of surfaces, each extremely close to one or the other of two merging surfaces.”

Edward Norton Lorenz (1917–2008) American mathematician and meteorologist

Lorenz (1963) "Deterministic nonperiodic flow", in: J. Atmos. Sci. 20, 130–141. cited in: T.N. Palmer (2008) " Edward Norton Lorenz. 23 May 1917 −− 16 April 2008 http://rsbm.royalsocietypublishing.org/content/55/139.full.pdf" in: Biogr. Mems Fell. R. Soc. 2009 55, 139-155

Related topics