Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
Preferencje help
Polish version English version Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Cover of the book
Tytuł książki

Some contributions to the differential geometry of submanifolds

Autorzy

Barbara Opozda

Seria

Rozprawy Matematyczne tom/nr w serii: 322 wydano: 1992

Zawartość

Pełne teksty: http://matwbn.icm.edu.pl/ksiazki/rm/rm322/rm32201.pdf [zdalny]

Warianty tytułu

Abstrakty

EN
CONTENTS
I. 1. Introduction..................................................................................................................................................................5
   2. Preliminaries..............................................................................................................................................................11
   3. On Simon’s conjecture..............................................................................................................................................13
II. Pinching theorems for submanifolds of the nearly Kähler 6-sphere..............................................................................16
   1. The nearly Kähler structure on S⁶(1)........................................................................................................................16
   2. 3-dimensional totally real submanifolds of S⁶............................................................................................................18
   3. Totally real surfaces in S⁶..........................................................................................................................................27
III. Surfaces in complex and Sasakian space forms with parallel mean curvature vector...................................................31
   1. Totally real surfaces in Kähler manifolds...................................................................................................................31
   2. Surfaces of genus 0 with parallel mean curvature vector..........................................................................................34
   3. Reduction theorems..................................................................................................................................................51
   4. Surfaces of genus 0, C-totally real immersed in Sasakian space forms with parallel mean curvature vector............56
References......................................................................................................................................................................63

Słowa kluczowe

Tematy

Kategoryzacja MSC:

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 322

Liczba stron

64

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CCCXXII

Daty

wydano
1992
otrzymano
1987-12-24
poprawiono
1992-04-01

Twórcy

autor
Barbara Opozda
  • Institute of Mathematics Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

Bibliografia

  • [A] K. Abe, Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions, Tôhoku Math. J. 25 (1973), 425-444.
  • [Ba] J. L. M. Barbosa, An extrinsic rigidity theorem for minimal immersions from $S^2$ into $S^n$, J. Differential Geom. 14 (1978), 355-368.
  • [BKSS] K. Benko, M. Kothe, K. D. Semler and U. Simon, Eigenvalues of the Laplacian and curvature, Colloq. Math. 42 (1979), 19-31.
  • [B] D. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer, Berlin 1976.
  • [BJRW] J. Bolton, G. R. Jensen, M. Rigoli and L. M. Woodward, On conformal minimal immersions of $S^2$ into $ℂP^n$, Math. Ann. 279 (1988), 599-620.
  • [B-W] J. Bolton and L. M. Woodward, On the Simon conjecture for minimal immersions with $S^1$-symmetry, Math. Z. 200 (1988), 111-121.
  • [Bo] O. Borůvka, Sur les surfaces représentées par les fonctions sphériques de première espèce, J. Math. Pures Appl. 12 (1933), 337-383.
  • [Br] R. Bryant, Minimal surfaces of constant curvature in $S^n$, Trans. Amer. Math. Soc. 290 (1985), 259-271.
  • [C]₁ E. Calabi, Isometric imbedding of complex manifolds, Ann. of Math. 58 (1953), 1-23.
  • [C]₂ E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc. 87 (1958), 407-438.
  • [C]₃ E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom. 1 (1967), 111-125.
  • [Ce] T. E. Cecil, Geometric applications of critical point theory to submanifolds of complex projective space, Nagoya Math. J. 55 (1974), 5-31.
  • [Ch]₁ B.-Y. Chen, On the surface with parallel mean curvature vector, Indiana Univ. Math. J. 22 (1973), 655-666.
  • [Ch]₂ B.-Y. Chen, Geometry of Submanifolds, Pure Appl. Math. 22, Marcel Dekker, New York 1973.
  • [ChHL] B.-Y. Chen, C. S. Houh and H.-S. Lue, Totally real submanifolds, J. Differential Geom. 12 (1977), 473-480.
  • [Ch-O] B.-Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974), 257-266.
  • [Che]₁ S. S. Chern, On the minimal immersions of the two-sphere in a space of constant curvature, in: Problems in Analysis, Princeton Univ. Press, Princeton, N.J., 1970, 27-40.
  • [Che]₂ S. S. Chern, On surfaces of constant mean curvature in a three-dimensional space of constant curvature, in: Lecture Notes in Math. 1007, Springer, Berlin 1983, 104-108.
  • [D] F. Dillen, Minimal immersions of surfaces into spheres, Arch. Math. (Basel) 49 (1987), 94-96.
  • [DOVV]₁ F. Dillen, B. Opozda, L. Verstraelen and L. Vrancken, On almost complex surfaces of the nearly Kaehler 6-sphere, I, in: Collection of Scientific Papers, Faculty of Science, Univ. of Kragujevac, 8 (1987), 5-13.
  • [DOVV]₂ F. Dillen, B. Opozda, L. Verstraelen and L. Vrancken, On totally real 3-dimensional submanifolds of the nearly Kaehler 6-sphere, Proc. Amer. Math. Soc. 99 (1987), 741-749.
  • [DOVV]₃ F. Dillen, B. Opozda, L. Verstraelen and L. Vrancken, On totally real surfaces of the nearly Kaehler 6-sphere, Geom. Dedicata 27 (1988), 325-334.
  • [DVV] F. Dillen, L. Verstraelen and L. Vrancken, On almost complex surfaces of the nearly Kaehler 6-sphere II, Kodai Math. J. 10 (3) (1987), 261-271.
  • [Eh] Ch. Ehresmann, Sur les variétés presque complexes, in: Proc. of the Internat. Congress of Mathematicians, 1950, Vol. II, Amer. Math. Soc., 1952, 412-419.
  • [E]₁ N. Ejiri, Totally real submanifolds in a 6-sphere, Proc. Amer. Math. Soc. 83 (1981), 759-763.
  • [E]₂ N. Ejiri, Equivariant minimal immersions of $S^2$ into $S^2m(1)$, Trans. Amer. Math. Soc. 297 (1) (1986), 105-124.
  • [Er] J. Erbacher, Reduction of codimension of an isometric immersion, J. Differential Geom. 5 (1971), 333-340.
  • [Fe] D. Ferus, Symmetric submanifolds of Euclidean space, Math. Ann. 247 (1980), 81-93.
  • [Fr] A. Fröhlicher, Zur Differentialgeometrie der komplexen Strukturen, Math. Ann. 129 (1955), 50-95.
  • [F-I] F. Fukami and S. Ishihara, Almost Hermitian structure on $S^6$, Tôhoku Math. J. 7 (1955), 151-156.
  • [G] A. Gray, Almost complex submanifolds of the six sphere, Proc. Amer. Math. Soc. 20 (1969), 277-279.
  • [G-R] I. V. Guadelupe and L. Rodriguez, Normal curvature of surfaces in space forms, Pacific J. Math. 106 (1983), 95-103.
  • [H] H. Hopf, Über Flächen mit einer Relation zwischen den Hauptkrümmungen, Math. Nachr. 4 (1950/51), 232-249.
  • [I] T. Itoh, A characterization of the generalized Veronese surfaces, Proc. Amer. Math. Soc. 104 (2) (1988), 571-576.
  • [J-R] G. Jensen and M. Rigoli, Minimal surfaces in spheres, Rend. Mat. Fis. Politec. Torino, Special volume, (1983), 75-98.
  • [K-Y] M. Kon and K. Yano, CR Submanifolds of Kaehlerian and Sasakian manifolds, Progr. Math. 30, Birkhäuser, Boston 1983.
  • [K-S] M. Kozlowski and U. Simon, Minimal immersions of 2-manifolds into spheres, Math. Z. 186 (1984), 377-382.
  • [L]₁ H. B. Lawson, Local rigidity theorem for minimal hypersurfaces, Ann. of Math. 89 (2) (1969), 187-197.
  • [L]₂ H. B. Lawson, The Riemannian geometry of holomorphic curves, in: Carolina Conference Proc. (1970), 45-62.
  • [M] K. Mashimo, Homogeneous totally real submanifolds of $S^6$, Tsukuba J. Math. 9 (1985), 185-202.
  • [MRU] S. Montiel, A. Ros and F. Urbano, Curvature pinching and eigenvalue rigidity for minimal submanifolds, Math. Z. 191 (1986), 537-548.
  • [N] H. Naitoh, Parallel submanifolds of complex space forms I, II, Nagoya Math. J. 90 (1983), 85-117, 91 (1983), 119-149.
  • [N-T] H. Naitoh and N. Takeuchi, Totally real submanifolds and symmetric bounded domains, Osaka J. Math. 19 (1982), 717-731.
  • [N-Ta] H. Nakagawa and R. Takagi, On locally symmetric Kaehler submanifolds in a complex projective space, J. Math. Soc. Japan 28 (1976), 638-667.
  • [O] T. Ogata, Minimal surfaces in a sphere with Gaussian curvature less than 1/6, Tôhoku Math. J. 37 (1985), 553-560.
  • [Og] K. Ogiue, Differential geometry of Kaehler submanifolds, Adv. in Math. 13 (1974), 73-114.
  • [Op]₁ B. Opozda, Totally real submanifolds of $S^6(1)$ with parallel second fundamental form, Suppl. Rend. Circ. Mat. Palermo 14 (1987), 247-253.
  • [Op]₂ B. Opozda, On totally real surfaces with parallel mean curvature vector, Bull. Soc. Math. Belg. Sér. B 40 (2) (1988), 207-244.
  • [R]₁ A. Ros, Positively curved Kaehler submanifolds, Proc. Amer. Math. Soc. 93 (1985), 329-331.
  • [R]₂ A. Ros, A characterization of seven compact Kaehler submanifolds by holomorphic pinching, Ann. of Math. 121 (1985), 377-382.
  • [R]₃ A. Ros, Kaehler submanifolds in the complex projective space, preprint.
  • [R-V] A. Ros and L. Verstraelen, On a conjecture of K. Ogiue, J. Differential Geom. 19 (1984), 561-566.
  • [Ru] E. A. Ruh, Minimal immersions of 2-spheres in $S^4$, Proc. Amer. Math. Soc. 28 (1971), 219-222.
  • [S] K. Sekigawa, Almost complex submanifolds of a 6-dimensional sphere, Kodai Math. J. 6 (1983), 174-185.
  • [Si] J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. 88 (1968), 62-105.
  • [S-Y] Y.-T. Siu and S.-T. Yau, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), 189-204.
  • [T] S. Tanno, Sasakian manifolds with constant ϕ-holomorphic sectional curvature, Tôhoku Math. J. 21 (1969), 501-507.
  • [U]₁ F. Urbano, Totally real minimal submanifolds of a complex projective space, Proc. Amer. Math. Soc. 93 (1985), 332-334.
  • [U]₂ F. Urbano, Nonnegatively curved totally real submanifolds, Math. Ann. 273 (1986), 345-348.
  • [W] N. Wallach, Extension of locally defined minimal immersions into spheres, Arch. Math. (Basel) 21 (1970), 210-213.
  • [We] H. C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), 193-243.
  • [YKM] S. Yamaguchi, M. Kon and Y. Miyahara, A theorem on C-totally real minimal surfaces, Proc. Amer. Math. Soc. 54 (1976), 276-280.
  • [Y] S.-T. Yau, Submanifolds with constant mean curvature I, Amer. J. Math. 96 (1974), 346-366.

Języki publikacji

EN

Uwagi

1991 Mathematics Subject Classification: Primary 53C20; Secondary 53A10.

Identyfikator YADDA

bwmeta1.element.dl-catalog-ff798d5e-67b2-452a-8539-d627f8259f47

Identyfikatory

ISBN
83-85116-63-X
ISSN
0012-3862

Kolekcja

DML-PL
Zawartość książki

rozwiń roczniki

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.