December  2018, 25: 87-95. doi: 10.3934/era.2018.25.009

On the embeddings of the Riemann sphere with nonnegative normal bundles

R. Pantilie, Institutul de Matematicǎ "Simion Stoilow" al Academiei Române, C.P. 1-764, 014700, Bucureşti, România

Received  May 30, 2018 Revised  November 23, 2018 Published  January 2019

Fund Project: This work is supported by a grant of the Ministery of Research and Innovation CNCS-UEFISCDI, project no. PN-Ⅲ-P4-ID-PCE-2016-0019, within PNCDI Ⅲ

We describe the (complex) quaternionic geometry encoded by the embeddings of the Riemann sphere with nonnegative normal bundles.

Citation: Radu Pantilie. On the embeddings of the Riemann sphere with nonnegative normal bundles. Electronic Research Announcements, 2018, 25: 87-95. doi: 10.3934/era.2018.25.009
References:
[1]

M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 85 (1957), 181-207. doi: 10.1090/S0002-9947-1957-0086359-5.

[2]

M. F. AtiyahN. J. Hitchin and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, 362 (1978), 425-461. doi: 10.1098/rspa.1978.0143.

[3]

T. N. Bailey and M. G. Eastwood, Complex paraconformal manifolds – their differential geometry and twistor theory, Forum Math., 3 (1991), 61-103. doi: 10.1515/form.1991.3.61.

[4]

E. Bonan, Sur les $G$-structures de type quaternionien, Cahiers Topologie Géom. Différentielle, 9 (1967), 389-461.

[5]

R. L. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory, in Complex Geometry and Lie Theory (Sundance, UT, 1989), Proc. Sympos. Pure Math., 53, Amer. Math. Soc., Providence, RI, 1991, 33–88. doi: 10.1090/pspum/053/1141197.

[6]

J. B. Carrell, A remark on the Grothendieck residue map, Proc. Amer. Math. Soc., 70 (1978), 43-48. doi: 10.1090/S0002-9939-1978-0492408-1.

[7]

Q.-S. Chi and L. J. Schwachhöfer, Exotic holonomy on moduli spaces of rational curves, Differential Geom. Appl., 8 (1998), 105-134. doi: 10.1016/S0926-2245(97)00019-3.

[8]

W. Fulton and J. Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9.

[9]

H. Grauert and R. Remmert, Coherent Analytic Sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 265, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69582-7.

[10]

M. Gualtieri, Generalized complex geometry, Ann. of Math. (2), 174 (2011), 75–123. doi: 10.4007/annals.2011.174.1.3.

[11]

N. J. Hitchin, Complex manifolds and Einstein's equations, in Twistor Geometry and Nonlinear Systems (Primorsko, 1980), Lecture Notes in Math., 970, Springer, Berlin-New York, 1982, 73–99.

[12]

P. Ionescu, Birational geometry of rationally connected manifolds via quasi-lines, in Projective Varieties With Unexpected Properties, Walter de Gruyter, Berlin, 2005,317–335.

[13]

D. Joyce, Compact hypercomplex and quaternionic manifolds, J. Differential Geom., 35 (1992), 743-761. doi: 10.4310/jdg/1214448266.

[14]

K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. (2), 75 (1962), 146–162. doi: 10.2307/1970424.

[15]

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. Ⅰ, Ⅱ, Ann. of Math. (2), 67 (1958), 328–466. doi: 10.2307/1970009.

[16]

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. Ⅲ. Stability theorems for complex structures, Ann. of Math. (2), 71 (1960), 43–76. doi: 10.2307/1969879.

[17]

C. R. LeBrun and S. Salamon, Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math., 118 (1994), 109-132. doi: 10.1007/BF01231528.

[18]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147. doi: 10.1112/blms/27.2.97.

[19]

S. MarchiafavaL. Ornea and R. Pantilie, Twistor Theory for CR quaternionic manifolds and related structures, Monatsh. Math., 167 (2012), 531-545. doi: 10.1007/s00605-011-0326-0.

[20]

S. Marchiafava and R. Pantilie, Twistor theory for co-CR quaternionic manifolds and related structures, Israel J. Math., 195 (2013), 347-371. doi: 10.1007/s11856-013-0001-3.

[21]

S. Merkulov and L. Schwachhöfer, Classification of irreducible holonomies of torsion-free affine connections, Ann. of Math. (2), 150 (1999), 77–149. doi: 10.2307/121098.

[22]

J. Morrow and H. Rossi, Submanifolds of $\mathbb{P}^N$ with splitting normal bundle sequence are linear, Math. Ann., 234 (1978), 253-261. doi: 10.1007/BF01420647.

[23]

M. S. Narasimhan, Deformations of complex structures and holomorphic vector bundles, in Complex Analysis, Proc. Summer School (Trieste, 1980) (ed. J. Eells), Lecture Notes in Math., Springer, 1982,196–209. doi: 10.1007/BFb0061878.

[24]

R. Pantilie, On the classification of the real vector subspaces of a quaternionic vector space, Proc. Edinb. Math. Soc. (2), 56 (2013), 615–622. doi: 10.1017/S0013091513000011.

[25]

R. Pantilie, On the twistor space of a (co-)CR quaternionic manifold, New York J. Math., 20 (2014), 959-971.

[26]

R. Pantilie, On the integrability of co-CR quaternionic structures, New York J. Math., 22 (2016), 1-20.

[27]

D. Quillen, Quaternionic algebra and sheaves on the Riemann sphere, Quart. J. Math. Oxford Ser. (2), 49 (1998), 163–198. doi: 10.1093/qmathj/49.2.163.

[28]

S. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4), 19 (1986), 31–55. doi: 10.24033/asens.1503.

show all references

References:
[1]

M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 85 (1957), 181-207. doi: 10.1090/S0002-9947-1957-0086359-5.

[2]

M. F. AtiyahN. J. Hitchin and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, 362 (1978), 425-461. doi: 10.1098/rspa.1978.0143.

[3]

T. N. Bailey and M. G. Eastwood, Complex paraconformal manifolds – their differential geometry and twistor theory, Forum Math., 3 (1991), 61-103. doi: 10.1515/form.1991.3.61.

[4]

E. Bonan, Sur les $G$-structures de type quaternionien, Cahiers Topologie Géom. Différentielle, 9 (1967), 389-461.

[5]

R. L. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory, in Complex Geometry and Lie Theory (Sundance, UT, 1989), Proc. Sympos. Pure Math., 53, Amer. Math. Soc., Providence, RI, 1991, 33–88. doi: 10.1090/pspum/053/1141197.

[6]

J. B. Carrell, A remark on the Grothendieck residue map, Proc. Amer. Math. Soc., 70 (1978), 43-48. doi: 10.1090/S0002-9939-1978-0492408-1.

[7]

Q.-S. Chi and L. J. Schwachhöfer, Exotic holonomy on moduli spaces of rational curves, Differential Geom. Appl., 8 (1998), 105-134. doi: 10.1016/S0926-2245(97)00019-3.

[8]

W. Fulton and J. Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9.

[9]

H. Grauert and R. Remmert, Coherent Analytic Sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 265, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69582-7.

[10]

M. Gualtieri, Generalized complex geometry, Ann. of Math. (2), 174 (2011), 75–123. doi: 10.4007/annals.2011.174.1.3.

[11]

N. J. Hitchin, Complex manifolds and Einstein's equations, in Twistor Geometry and Nonlinear Systems (Primorsko, 1980), Lecture Notes in Math., 970, Springer, Berlin-New York, 1982, 73–99.

[12]

P. Ionescu, Birational geometry of rationally connected manifolds via quasi-lines, in Projective Varieties With Unexpected Properties, Walter de Gruyter, Berlin, 2005,317–335.

[13]

D. Joyce, Compact hypercomplex and quaternionic manifolds, J. Differential Geom., 35 (1992), 743-761. doi: 10.4310/jdg/1214448266.

[14]

K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. (2), 75 (1962), 146–162. doi: 10.2307/1970424.

[15]

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. Ⅰ, Ⅱ, Ann. of Math. (2), 67 (1958), 328–466. doi: 10.2307/1970009.

[16]

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. Ⅲ. Stability theorems for complex structures, Ann. of Math. (2), 71 (1960), 43–76. doi: 10.2307/1969879.

[17]

C. R. LeBrun and S. Salamon, Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math., 118 (1994), 109-132. doi: 10.1007/BF01231528.

[18]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147. doi: 10.1112/blms/27.2.97.

[19]

S. MarchiafavaL. Ornea and R. Pantilie, Twistor Theory for CR quaternionic manifolds and related structures, Monatsh. Math., 167 (2012), 531-545. doi: 10.1007/s00605-011-0326-0.

[20]

S. Marchiafava and R. Pantilie, Twistor theory for co-CR quaternionic manifolds and related structures, Israel J. Math., 195 (2013), 347-371. doi: 10.1007/s11856-013-0001-3.

[21]

S. Merkulov and L. Schwachhöfer, Classification of irreducible holonomies of torsion-free affine connections, Ann. of Math. (2), 150 (1999), 77–149. doi: 10.2307/121098.

[22]

J. Morrow and H. Rossi, Submanifolds of $\mathbb{P}^N$ with splitting normal bundle sequence are linear, Math. Ann., 234 (1978), 253-261. doi: 10.1007/BF01420647.

[23]

M. S. Narasimhan, Deformations of complex structures and holomorphic vector bundles, in Complex Analysis, Proc. Summer School (Trieste, 1980) (ed. J. Eells), Lecture Notes in Math., Springer, 1982,196–209. doi: 10.1007/BFb0061878.

[24]

R. Pantilie, On the classification of the real vector subspaces of a quaternionic vector space, Proc. Edinb. Math. Soc. (2), 56 (2013), 615–622. doi: 10.1017/S0013091513000011.

[25]

R. Pantilie, On the twistor space of a (co-)CR quaternionic manifold, New York J. Math., 20 (2014), 959-971.

[26]

R. Pantilie, On the integrability of co-CR quaternionic structures, New York J. Math., 22 (2016), 1-20.

[27]

D. Quillen, Quaternionic algebra and sheaves on the Riemann sphere, Quart. J. Math. Oxford Ser. (2), 49 (1998), 163–198. doi: 10.1093/qmathj/49.2.163.

[28]

S. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4), 19 (1986), 31–55. doi: 10.24033/asens.1503.

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