|
Manifolds with preassigned curvature—a survey
Author(s):
Herman
Gluck
Journal:
Bull. Amer. Math. Soc.
81
(1975),
313-329.
MSC (1970):
Primary 53-02, 53C20, 53C45
MathSciNet review:
0367861
Retrieve article in:
PDF
References |
Similar articles |
Additional information
References:
- 1.
- A. D. Alexandrov, Zur Theorie gemischter Volumina konvexer Körper. I, II, Mat. Sb. 2 (1937), 947-972, 1205-1238.
- 2.
- A. D. Alexandrov, Zur Theorie gemischter Volumina. III, Mat. Sb. 3 (1938).
- 3.
- A. D. Alexandrov, Über die Oberflächenfunktion eines konvexen Körpers. (Bemerkung zur Arbeit "Zur Theorie der gemischten Volumina von konvexen Körpern", Mat. Sb. 6 (48) (1939), 167-174. (Russian) MR 1, 265.
- 4.
- A. D. Alexandrov, Smoothness of the convex surface of bounded Gaussian curvature, C.R. (Dokl.) Acad. Sci. URSS 36 (1942), 195-199. MR 4, 169. MR 7626
- 5.
- A. D. Alexandrov, Intrinsic geometry of convex surfaces, OGIZ, Moscow, 1948; German transl., Akademie-Verlag, Berlin, 1955. MR 10, 619; 17, 74. MR 29518
- 6.
- A. D. Alexandrov, Convex polyhedra, GITTL, Moscow, 1950; German transl., Akademie-Verlag, Berlin, 1958. MR 12, 732; 19, 1192. MR 2127379
- 7.
- T. F. Banchoff, Critical points and curvature for embedded polyhedra, J. Differential Geometry 1 (1967), 245-256. MR 37 #921. MR 225327
- 8.
- Melvyn S. Berger, Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, J. Differential Geometry 5 (1971), 325-332. MR 45 #4329. MR 295261
- 9.
- T. Bonnesen and W. Fenchel, Theorie der Konvexen Körper, Julius Springer, Berlin, 1934. MR 344997
- 10.
- O. Bonnet, J. École Polytechnique 19 (1848), 1-146.
- 11.
- J.-P. Bourguignon, Thesis, Univ. of Paris VII, 1974.
- 12.
- A. Cauchy, Sur les polygones et polyedres, Second Memoire, J. École Polytechnique 9 (1813), 87.
- 13.
- S.-S. Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc. 6 (1955), 771-782. MR 17, 657. MR 74856
- 14.
- A. Fisher and J. Marsden, The manifold of Riemannian metrics with prescribed scalar curvature (to appear).
- 15.
- C. F. Gauss, Disquisitiones generales circa superficies curvas, 1827; English translation: General investigations on curved surfaces, Princeton, 1902.
- 16.
- H. Gluck, Deformations of normal vector fields and the generalized Minkowski problem, Bull. Amer. Math. Soc. 77 (1971), 1106-1110. MR 44 #2179. MR 284955
- 17.
- H. Gluck, The generalized Minkowski problem in differential geometry in the large, Ann. of Math. (2) 96 (1972), 245-276. MR 46 #8132. MR 309021
- 18.
- H. Gluck, The converse to the Four-Vertex Theorem, Enseignement Math. 17 (1971), 295-309. MR 344998
- 19.
- H. Gluck, K. Krigelman and D. Singer, The converse to the Gauss-Bonnet Theorem in PL, J. Differential Geometry 9 (1974), 601-616. MR 390962
- 20.
- J. L. Kazdan and F. W. Warner, Integrability conditions for Δ u=k-Ke, Bull. Amer. Math. Soc. 77 (1971), 819-823. MR 43 #8026. MR 282314
- 21.
- J. L. Kazdan and F. W. Warner, Curvature functions for 2-manifolds with negative Euler characteristic, Bull. Amer. Math. Soc. 78 (1972), 570-574. MR 320947
- 22.
- J. L. Kazdan and F. W. Warner, Curvature functions for 2-manifolds, Proc. Sympos. Pure Math. vol. 23, Amer. Math. Soc., Providence, R.I., 1973, pp. 387-392. MR 343207
- 23.
- J. L. Kazdan and F. W. Warner, Surfaces of revolution with monotonic increasing curvature, and an application to the equation Δu=l—Ke, Proc. Amer. Math. Soc. 32 (1972), 139-141.
- 24.
- J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. (2) 99 (1974), 14-47. MR 343205
- 25.
- J. L. Kazdan and F. W. Warner, Curvature functions for open 2-manifolds, Ann. of Math. (2) 99 (1974), 203-219. MR 343206
- 26.
- J. L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry (to appear). MR 365409
- 27.
- J. L. Kazdan and F. W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian scalar curvatures, Ann. of Math. (to appear). MR 375153
- 28.
- J. L. Kazdan and F. W. Warner, A direct approach to the determination of Gaussian and scalar curvature functions (to appear). MR 375154
- 29.
- D. Koutroufiotis, On Gaussian curvature and conformal mapping, J. Differential Geometry 7 (1972), 481-490. MR 328835
- 30.
- H. Lewy, On the existence of a closed surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci. U.S.A. 24 (1938), 104-106.
- 31.
- L. A. Ljusternik, Convex figures and polyhedra, GITTL, Moscow, 1956; English transl., Dover, New York, 1963. MR 19, 57; 28 #4427. MR 161219
- 32.
- H. Minkowski, Allgemeine Lehrsätze über konvexe Polyeder, Nachr. Ges. Wiss., Göttingen 1897, 198-219.
- 33.
- H. Minkowski, Volumen unci Oberflache, Math. Ann. 57 (1903), 447-495. MR 1511220
- 34.
- J. Moser, On a nonlinear problem in differential geometry, Dynamical Systems, Academic Press, New York, 1973, pp. 273-280. MR 339258
- 35.
- J. Moser, A sharp form of an inequality of N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077-1092. MR 46 #662. MR 301504
- 36.
- J. F. Nash, Jr., The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20-63. MR 17, 782. MR 75639
- 37.
- L. Nirenberg, The Weyl and Minkowski Problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337-394. MR 15, 347. MR 58265
- 38.
- A. V. Pogorelov, Regularity of a convex surface with given Gaussian curvature, Mat. Sb. 31 (73) (1952), 88-103. (Russian) MR 14, 679. MR 52807
- 39.
- A. V. Pogorelov, On a regular solution of the n-dimensional Minkowski problem, Dokl. Akad. Nauk SSSR 199 (1971), 785-788=Soviet Math. Dokl. 12 (1971), 1192-1196. MR 44 #2180. MR 284956
- 40.
- H. Poincaré, Les fonctions fuchsiennes et l'equation Δ u=e, J. Math. (5) 4 (1898), 137-230.
- 41.
- D. Singer, Preassigning curvature on the two-sphere, J. Differential Geometry 9 (1974), 633-638. MR 394697
- 42.
- D. Stone, Geodesics in PL manifolds (to appear).
- 43.
- H. Weyl, Über die bestimmung einer geschlossenen konvexen flache durch ihr linienelement, Vierteljahrsschrift der naturforschen der Gesellschaft Zürich 61 (1916), 40-72.
Similar Articles:
Retrieve articles in Bulletin of the American Mathematical Society
with MSC
(1970):
53-02, 53C20, 53C45
Retrieve articles in all Journals with MSC
(1970):
53-02, 53C20, 53C45
Additional Information:
DOI:
10.1090/S0002-9904-1975-13740-2
PII:
S 0002-9904(1975)13740-2
|
