Riemannian connection on a surface

Riemannian connection on a surface

:"For the classical approach to the geometry of surfaces, see Differential geometry of surfaces."

In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Elie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form . These concepts were put in their final form using the language of principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.

Historical overview

[
Tullio Levi-Civita (1873-1941)]
[
thumb|right|180px|Élie Cartan (1869-1951)]
[
thumb|right|130px|Hermann Weyl (1885-1955)] After the classical work of Gauss on the differential geometry of surfaces [
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*citation|last=Nelson|first= Edward|authorlink=Edward Nelson|title=Topics in dynamics — I: Flows| series=Mathematical Notes|publisher= Princeton University Press|year= 1969
*citation|last= O'Neill|first=Barrett | title = Elementary Differential Geometry | publisher=Academic Press | year=1997 | id = ISBN 0-12-526745-2
*citation|last=Poznjak|first=E.G.|title=Isometric imbedding of two-dimensional Riemannian metrics in Euclidean spaces
journal=Russian Math. Surveys|volume= 28 |year=1973|pages= 47-77
*citation|first=Andrew|last=Pressley|title=Elementary Differential Geometry|series=Springer Undergraduate Mathematics Series|year=2001|publisher=Springer-Verlag|id= ISBN 1852331526
*citation|last=Sasaki|first=Shigeo|title=On the differential geometry of tangent bundles of Riemannian manifolds|journal=Tôhoku Math. J.|volume=10| year= 1958| pages=338-354
*citation|title=Differential Geometry: Cartan's Generalization of Klein's Erlangen Program|first= Richard W.|last= Sharpe|publisher=Springer-Verlag| year =1997|id=ISBN 0387947329
*citation | last1=Singer|first1=Isadore M.|authorlink1=Isadore Singer| last2= Thorpe|first2=John A. | title= Lecture Notes on Elementary Topology and Geometry | publisher=Springer-Verlag | year=1967 | id=ISBN 0-387-90202-3
*citation|last=Sternberg|first=Shlomo|year=1964|title=Lectures on differential geometry|publisher= Prentice-Hall
*citation|title=Lectures on classical differential geometry: Second Edition|first=Dirk Jan |last=Struik|authorlink=Dirk Struik|year= 1988|publisher= Dover
id=ISBN 0486656098
*citation|title=Differential Geometry of Curves and Surfaces: A Concise Guide| first= Victor A.|last=Toponogov| authorlink= Victor Andreevich Toponogov|date=2005|publisher=Springer-Verlag|id=ISBN 0817643842
*citation|last=Valiron|first=Georges|title=The Classical Differential Geometry of Curves and Surfaces|year =1986|publisher= Math Sci Press|id=ISBN0915692392 [http://books.google.com/books?id=IQXstKvWsHMC&printsec=frontcover&dq=valiron+surfaces&source=gbs_summary_r&cad=0 Full text of book]
*citation|title=Lie Groups, Lie Algebras, and Their Representations|first= V. S. |last=Varadarajan|publisher= Springer-Verlag
year= 1984|id=ISBN 0387909699
*citation|last=Wilson|first=Pelham|title=Curved Space: From Classical Geometries to Elementary Differential Geometry|publisher=Cambridge University Press|year=2008|id=ISBN 978-0-521-71390-0

External links

*citation|first=Eugenio|last=Calabi|authorlink=Eugenio Calabi|url=http://www.seas.upenn.edu/~cis70005/cis700sl6pdf.pdf|title=Basics of the differential geometry of surfaces, Part I|publisher=University of Pennsylvania
*citation|first=Eugenio|last=Calabi|authorlink=Eugenio Calabi|url=http://www.seas.upenn.edu/~cis70005/cis700sl7pdf.pdf|title=Basics of the differential geometry of surfaces, Part II|publisher=University of Pennsylvania
*citation|first=Eugenio|last=Calabi|authorlink=Eugenio Calabi|url=http://www.seas.upenn.edu/~cis70005/cis700sl8pdf.pdf|title=Basics of the differential geometry of surfaces, Part III|publisher=University of Pennsylvania
*citation|first=Eugenio|last=Calabi|authorlink=Eugenio Calabi|url=http://www.seas.upenn.edu/~cis70005/cis700sl9pdf.pdf|title=Basics of the differential geometry of surfaces, Part IV|publisher=University of Pennsylvania
*citation|first=Eugenio|last=Calabi|authorlink=Eugenio Calabi|url=http://www.seas.upenn.edu/~cis70005/cis700sl10pdf.pdf|title=Basics of the differential geometry of surfaces, Part V|publisher=University of Pennsylvania
*citation|first=Eugenio|last=Calabi|authorlink=Eugenio Calabi|url=http://www.seas.upenn.edu/~cis70005/cis700sl11pdf.pdf|title=Basics of the differential geometry of surfaces, Parts VI and VII|publisher=University of Pennsylvania
*citation|first=Eugenio|last=Calabi|authorlink=Eugenio Calabi|url=http://www.seas.upenn.edu/~cis70005/cis700sl12.ps|title=Basics of the differential geometry of surfaces, Part VIII|publisher=University of Pennsylvania


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