Riemannian connection on a surface (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Riemannian connection on a surface" in English language version.

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Eisenhart 2004 Eisenhart, Luther P. (2004), A Treatise on the Differential Geometry of Curves and Surfaces, Dover, ISBN 0486438201 Full 1909 text (now out of copyright)
Arnold 1989, pp. 301–306, Appendix I. Arnold, V. I. (1989), Mathematical Methods of Classical Mechanics, Springer-Verlag, ISBN 0-387-96890-3
Arnold 1989, Appendix 2: Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids. Arnold, V. I. (1989), Mathematical Methods of Classical Mechanics, Springer-Verlag, ISBN 0-387-96890-3

doi.org

Levi-Civita 1917 Levi-Civita, Tullio (1917), "Nozione di parallelismo in una varietà qualunque", Rend. Circ. Mat. Palermo, 42 (1): 173–205, doi:10.1007/BF03014898, S2CID 122088291
Sasaki 1958 Sasaki, Shigeo (1958), "On the differential geometry of tangent bundles of Riemannian manifolds", Tôhoku Mathematical Journal, 10 (3): 338–354, doi:10.2748/tmj/1178244668
Kobayashi 1956 Kobayashi, Shochichi (1956), "Induced connections and imbedded Riemannian space", Nagoya Math. J., 10: 15–25, doi:10.1017/S0027763000000052, S2CID 251061784
The definition presented here is due essentially to Charles Ehresmann. However, it is different from, though related to, what is commonly called an Ehresmann connection. It is also different from, though related to, what is commonly called a Cartan connection. See Kobayashi (1957) and Sharpe (1997) for a survey of some of the various types of connections and the relations between them. Kobayashi, Shochichi (1957), "Theory of connections", Annali di Matematica Pura ed Applicata, Series 4, 43 (1): 119–194, doi:10.1007/BF02411907, S2CID 120972987, Sharpe, Richard W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, ISBN 0387947329
Klingenberg & Sasaki 1975 Klingenberg, Wilhelm; Sasaki, Shigeo (1975), "On the tangent sphere bundle of a 2-sphere", Tôhoku Mathematical Journal, 27 (1): 49–56, doi:10.2748/tmj/1178241033
A similar argument applies to the transitive action by conjugation of SU(2) on matrices F = 2Q − I with Q a rank one projection in M₂(C). This action is trivial on ± I, so passes to a transitive action of SO(3) with stabilizer subgroup SO(2), showing that these matrices provide another model for S². This is standard material in gauge theory on SU(2); see for example Narasimhan & Ramadas (1979). Narasimhan, M.S.; Ramadas, T. R. (1979), "Geometry of SU(2) gauge fields", Comm. Math. Phys., 67 (2): 121–136, Bibcode:1979CMaPh..67..121N, doi:10.1007/BF01221361, S2CID 118840198
Kobayashi 1956, Theorem II. Kobayashi, Shochichi (1956), "Induced connections and imbedded Riemannian space", Nagoya Math. J., 10: 15–25, doi:10.1017/S0027763000000052, S2CID 251061784

harvard.edu

ui.adsabs.harvard.edu

A similar argument applies to the transitive action by conjugation of SU(2) on matrices F = 2Q − I with Q a rank one projection in M₂(C). This action is trivial on ± I, so passes to a transitive action of SO(3) with stabilizer subgroup SO(2), showing that these matrices provide another model for S². This is standard material in gauge theory on SU(2); see for example Narasimhan & Ramadas (1979). Narasimhan, M.S.; Ramadas, T. R. (1979), "Geometry of SU(2) gauge fields", Comm. Math. Phys., 67 (2): 121–136, Bibcode:1979CMaPh..67..121N, doi:10.1007/BF01221361, S2CID 118840198

semanticscholar.org

api.semanticscholar.org

Levi-Civita 1917 Levi-Civita, Tullio (1917), "Nozione di parallelismo in una varietà qualunque", Rend. Circ. Mat. Palermo, 42 (1): 173–205, doi:10.1007/BF03014898, S2CID 122088291
Kobayashi 1956 Kobayashi, Shochichi (1956), "Induced connections and imbedded Riemannian space", Nagoya Math. J., 10: 15–25, doi:10.1017/S0027763000000052, S2CID 251061784
The definition presented here is due essentially to Charles Ehresmann. However, it is different from, though related to, what is commonly called an Ehresmann connection. It is also different from, though related to, what is commonly called a Cartan connection. See Kobayashi (1957) and Sharpe (1997) for a survey of some of the various types of connections and the relations between them. Kobayashi, Shochichi (1957), "Theory of connections", Annali di Matematica Pura ed Applicata, Series 4, 43 (1): 119–194, doi:10.1007/BF02411907, S2CID 120972987, Sharpe, Richard W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, ISBN 0387947329
A similar argument applies to the transitive action by conjugation of SU(2) on matrices F = 2Q − I with Q a rank one projection in M₂(C). This action is trivial on ± I, so passes to a transitive action of SO(3) with stabilizer subgroup SO(2), showing that these matrices provide another model for S². This is standard material in gauge theory on SU(2); see for example Narasimhan & Ramadas (1979). Narasimhan, M.S.; Ramadas, T. R. (1979), "Geometry of SU(2) gauge fields", Comm. Math. Phys., 67 (2): 121–136, Bibcode:1979CMaPh..67..121N, doi:10.1007/BF01221361, S2CID 118840198
Kobayashi 1956, Theorem II. Kobayashi, Shochichi (1956), "Induced connections and imbedded Riemannian space", Nagoya Math. J., 10: 15–25, doi:10.1017/S0027763000000052, S2CID 251061784

ucsd.edu

math.ucsd.edu

This standard treatment of parallel transport can be found for example in Driver (1995, p. 25). Driver, Bruce K. (1995), A primer on Riemannian geometry and stochastic analysis on path spaces (PDF), Lectures given at the E.T.H., Zurich

umich.edu

hti.umich.edu

Darboux 1890 Darboux, Gaston (1890), Leçons sur la théorie générale des surfaces, Gauthier-Villars Volume I, Volume II, Volume III, Volume IV

zenodo.org

Levi-Civita 1917 Levi-Civita, Tullio (1917), "Nozione di parallelismo in una varietà qualunque", Rend. Circ. Mat. Palermo, 42 (1): 173–205, doi:10.1007/BF03014898, S2CID 122088291