“The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves.”

— Shiing-Shen Chern

[Differential geometry, its past and its future, Actes, Congrès inter. math, 1970, 41–53, http://www.math.harvard.edu/~hirolee/pdfs/2014-fall-230a-icm1970-chern-differential-geometry.pdf]

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 main object of study in differential geometry is, at least for the moment, the differential manifolds, structures o…" by Shiing-Shen Chern?

Shiing-Shen Chern 6

mathematician (1911–2004), born in China and later acquirin… 1911–2004

Related quotes

“Not all the geometrical structures are "equal". It would seem that the riemannian and complex structures, with their contacts with other fields of mathematics and with their richness in results, should occupy a central position in differential geometry. A unifying idea is the notion of a G-structure, which is the modern version of an equivalence problem first emphasized and exploited in its various special cases by Elie Cartan.”

Shiing-Shen Chern (1911–2004) mathematician (1911–2004), born in China and later acquiring U.S. citizenship; made fundamental contributio…

“We must… maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds.”

Hans Reichenbach (1891–1953) American philosopher

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

“For Space, when the position of points is expressed by rectilinear co-ordinates, ds = \sqrt{ \sum (dx)^2 }; Space is therefore included in this simplest case. The next case in simplicity includes those manifoldnesses in which the line-element may be expressed as the fourth root of a quartic differential expression…. I restrict myself… to those manifoldnesses in which the line element is expressed as the square root of a quadric differential expression…. Manifoldnesses in which, as in the Plane and in Space, the line-element may be reduced to the form \sqrt{ \sum (dx)^2 }, are… only a particular case of the manifoldnesses to be here investigated; they require a special name, and therefore these manifoldnesses… I will call flat. In order now to review the true varieties of all the continua which may be represented in the assumed form, it is necessary to get rid of difficulties arising from the mode of representation, which is accomplished by choosing the variables in accordance with a certain principle.”

Bernhard Riemann (1826–1866) German mathematician

On the Hypotheses which lie at the Bases of Geometry (1873)

“Cartan developed a general scheme of infinitesimal geometry in which Klein's notions were applied to the tangent plane and not to the n-dimensional manifold M itself.”

Hermann Weyl (1885–1955) German mathematician

On the foundations of general infinitesimal geometry. Bull. Amer. Math. Soc. 35 (1929) 716–725 [10.1090/S0002-9904-1929-04812-2] (quote on p. 716)

“[The law of evolution states that] complexity increases in terms of differentiation and structure.”

Kenneth E. Boulding (1910–1993) British-American economist

Source: 1970s, Ecodynamics: A New Theory Of Societal Evolution, 1978, p. 10 as cited in P.P. Kandelaars (1999) Economic Models of Material-Product Chains for Environmental Policy Analysis. p. 13

“Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.”

Hans Reichenbach (1891–1953) American philosopher

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

“Don't sign your name
between worlds,

surmount
the manifold of meanings,

trust the tearstain,
learn to live.”

Paul Celan (1920–1970) Romanian poet and translator

Source: Glottal Stop

“The truth is that other systems of geometry are possible, yet after all, these other systems are not spaces but other methods of space measurements. There is one space only, though we may conceive of many different manifolds, which are contrivances or ideal constructions invented for the purpose of determining space.”

Paul Carus (1852–1919) American philosopher

Science, Vol. 18 (1903), p. 106, as reported in Memorabilia Mathematica; or, The Philomath's Quotation-Book https://archive.org/stream/memorabiliamathe00moriiala#page/81/mode/2up, (1914), by Robert Edouard Moritz, p. 352

“If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension.”

Bernhard Riemann (1826–1866) German mathematician

On the Hypotheses which lie at the Bases of Geometry (1873)

“The ultimate "causes of price" - to use a Classical term - lie deeply embedded in the psychology and techniques of mankind and his environment, and are as manifold as the sands of the sea. All economic analysis is an attempt to classify these manifold causes, to sort them out into categories of discourse that our limited minds can handle, and so to perceive the unity of structural relationship which both unites and separates the manifoldness. Our concepts of "" and "supply" are such broad categories. In whatever sense they are used, they are not ultimate determinants of anything, but they are convenient channels through which we can classify and describe the effects of the multitude of determinants of the system of economic magnitude.”

Kenneth E. Boulding (1910–1993) British-American economist

Kenneth Boulding (1944) " A Liquidity Preference Theory of Market Prices http://cas.umkc.edu/econ/economics/faculty/wray/631Wray/Week%207/Boulding.pdf". In: Economica, New Series, Vol. 11, No. 42 (May, 1944), pp. 55-63.
C. Brown (2003) " Toward a reconcilement of endogenous money and liquidity preference http://www.clt.astate.edu/crbrown/brownjpke.pdf" in: Journal of Post Keynesian Economics. Winter 2003–4, Vol. 26, No. 2. 323 commented on this article, saying: "Boulding (1944) argued that if liquidity preference were divorced from the "demand for money," the former could come into its own as a theory of financial asset pricing. According to this view, rising liquidity preference or a "wave of bearish sentiment" is manifest in a shift from certain asset categories, specifically, those that are characterized by high capital uncertainty (that is, uncertainty about the future value of the asset as a result of market revaluation) to assets such as commercial paper or giltedged securities."
1940s

Help us to complete the source, original and additional information

Shiing-Shen Chern 6

Related quotes

Related topics