The Schwarzian derivative and conformal mapping of Riemannian manifolds
Brad Osgood and Dennis Stowe
Source: Duke Math. J. Volume 67, Number 1 (1992), 57-99.
First Page PDF: View first page of article (PDF, 66 KB)Primary Subjects: 53C25
Secondary Subjects: 30C65, 53A30, 53B10
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077294272
Mathematical Reviews number (MathSciNet):
MR1174603
Zentralblatt MATH identifier:
0766.53034
Digital Object Identifier: doi:10.1215/S0012-7094-92-06704-4
References
[A] L. V. Ahlfors, Cross-ratios and Schwarzian derivatives in ${\bf R}\sp n$, Complex analysis, Birkhäuser, Basel, 1988, Articles Dedicated to Albert Pfluger on the Occasion of his 80th Birthday, pp. 1–15.
Mathematical Reviews (MathSciNet):
MR90a:30055
Zentralblatt MATH:
0675.30021
[BK] C. Barbance and Y. Kerbrat, Sur les transformations conformes des variétés d'Einstein, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 8, A391–A394.
Mathematical Reviews (MathSciNet):
MR57:13786
Zentralblatt MATH:
0379.53026
[Be] A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987.
Mathematical Reviews (MathSciNet):
MR88f:53087
Zentralblatt MATH:
0613.53001
[Br] H. W. Brinkmann, Einstein spaces which are mapped conformally on each other, Math. Ann. 94 (1925), 119–145.
Zentralblatt MATH:
51.0568.03
Mathematical Reviews (MathSciNet):
MR1512246
Digital Object Identifier: doi:10.1007/BF01208647
[Ca] K. Carne, The Schwarzian derivative for conformal maps, J. Reine Angew. Math. 408 (1990), 10–33.
Mathematical Reviews (MathSciNet):
MR91h:30040
Zentralblatt MATH:
0705.30010
[C] M. Chuaqui, The Schwarzian derivative in Riemannian geometry; univalence criteria and quasiconformal reflections, Ph.D. thesis, Stanford Univ., 1990.
[E] L. P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, N. J., 1949.
Mathematical Reviews (MathSciNet):
MR11,687g
Zentralblatt MATH:
0041.29403
[F] A. Fialkow, Conformal geodesics, Trans. Amer. Math. Soc. 45 (1939), no. 3, 443–473.
Mathematical Reviews (MathSciNet):
MR1501998
Zentralblatt MATH:
0021.06501
Digital Object Identifier: doi:10.2307/1990011
[G] F. W. Gehring, Spirals and the universal Teichmüller space, Acta Math. 141 (1978), no. 1-2, 99–113.
Mathematical Reviews (MathSciNet):
MR58:17076
Zentralblatt MATH:
0393.30015
[Ko] S. Kobayashi, A theorem on the affine transformation group of a Riemannian manifold, Nagoya Math. J. 9 (1955), 39–41.
Mathematical Reviews (MathSciNet):
MR17,892a
Zentralblatt MATH:
0067.14501
[K1] R. S. Kulkarni, Curvature structures and conformal transformations, J. Differential Geometry 4 (1970), 425–451.
Mathematical Reviews (MathSciNet):
MR44:2173
Zentralblatt MATH:
0206.24403
[K2] R. S. Kulkarni, Curvature and metric, Ann. of Math. (2) 91 (1970), 311–331.
Mathematical Reviews (MathSciNet):
MR41:2581
Zentralblatt MATH:
0191.19903
Digital Object Identifier: doi:10.2307/1970580
JSTOR: links.jstor.org
[L] O. Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987.
Mathematical Reviews (MathSciNet):
MR88f:30073
Zentralblatt MATH:
0606.30001
[MS] O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 2, 383–401.
Mathematical Reviews (MathSciNet):
MR81i:30039
Zentralblatt MATH:
0406.30013
[Na] T. Nagano, The conformal transformation on a space with parallel Ricci tensor. , J. Math. Soc. Japan 11 (1959), 10–14.
Mathematical Reviews (MathSciNet):
MR23:A1330
Zentralblatt MATH:
0089.17201
[N] Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551.
Mathematical Reviews (MathSciNet):
MR10,696e
Zentralblatt MATH:
0035.05104
[O] M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247–258.
Mathematical Reviews (MathSciNet):
MR46:2601
Zentralblatt MATH:
0236.53042
[O'N] B. O'Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983.
Mathematical Reviews (MathSciNet):
MR85f:53002
Zentralblatt MATH:
0531.53051
[OS1] B. Osgood and D. Stowe, A generalization of Nehari's univalence criterion, Comment. Math. Helv. 65 (1990), no. 2, 234–242.
Mathematical Reviews (MathSciNet):
MR92a:53015
Zentralblatt MATH:
0722.53029
[OS2] B. Osgood and D. Stowe, The Möbius connection in the bundle of conformal $2$-jets, preprint.
[Tk] N. Tanaka, Conformal connections and conformal transformations, Trans. Amer. Math. Soc. 92 (1959), 168–190.
Mathematical Reviews (MathSciNet):
MR23:A1331
Zentralblatt MATH:
0096.15603
Digital Object Identifier: doi:10.2307/1993174
[T1] Y. Tashiro, Remarks on a theorem concerning conformal transformations, Proc. Japan Acad. 35 (1959), 421–422.
Mathematical Reviews (MathSciNet):
MR22:11339
Zentralblatt MATH:
0101.14202
[T2] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251–275.
Mathematical Reviews (MathSciNet):
MR30:4229
Zentralblatt MATH:
0136.17701
Digital Object Identifier: doi:10.2307/1994206
[T] W. P. Thurston, Zippers and univalent functions, The Bieberbach conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr., vol. 21, Amer. Math. Soc., Providence, RI, 1986, Proceedings of a Symposium on the Occasion of its Proof, pp. 185–197.
Mathematical Reviews (MathSciNet):
MR88j:30040
[Yn]1 K Yano, Concircular geometry. I. Concircular transformations, Proc. Imp. Acad. Tokyo 16 (1940), 195–200.
Mathematical Reviews (MathSciNet):
MR2,165a
Zentralblatt MATH:
0024.08102
[Yn]2 K. Yano, Concircular geometry. II. Integrability conditions of $\rho\sb {\mu\nu}=\phi g\sb {\mu\nu}$, Proc. Imperial Acad. Tokyo 16 (1940), 354–360.
Mathematical Reviews (MathSciNet):
MR2,165b
Zentralblatt MATH:
0024.18403
[Yn]3 K. Yano, Concircular geometry III. Theory of curves, Proc. Imperial Acad. Tokyo 16 (1940), 442–448.
Mathematical Reviews (MathSciNet):
MR2,303d
Zentralblatt MATH:
0025.08503
[Yn]4 K. Yano, Concircular geometry IV. Theory of subspaces, Proc. Imperial Acad. Tokyo 16 (1940), 505–511.
Mathematical Reviews (MathSciNet):
MR2,303e
Zentralblatt MATH:
0025.08504
[Yn]5 K. Yano, Concircular geometry. V. Einstein spaces, Proc. Imperial Acad. Tokyo 18 (1942), 446–451.
Mathematical Reviews (MathSciNet):
MR7,330f
Zentralblatt MATH:
0060.38601
[Y] S. T. Yau, Remarks on conformal transformations, J. Differential Geometry 8 (1973), 369–381.
Mathematical Reviews (MathSciNet):
MR49:3770
Zentralblatt MATH:
0274.53047
Duke Mathematical Journal