Horizontal lift of symmetric connections to the bundle of volume forms 𝒱

Anna Gasior

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2010)

  • Volume: 54, Issue: 1
  • ISSN: 0365-1029

Abstract

top
In this paper we present the horizontal lift of a symmetric affine connection with respect to another affine connection to the bundle of volume forms  𝒱 and give formulas for its curvature tensor, Ricci tensor and the scalar curvature. Next, we give some properties of the horizontally lifted vector fields and certain infinitesimal transformations. At the end, we consider some substructures of a F ( 3 , 1 ) -structure on 𝒱 .

How to cite

top

Anna Gasior. "Horizontal lift of symmetric connections to the bundle of volume forms $\mathcal {V}$." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 54.1 (2010): null. <http://eudml.org/doc/289759>.

@article{AnnaGasior2010,
abstract = {In this paper we present the horizontal lift of a symmetric affine connection with respect to another affine connection to the bundle of volume forms $\mathcal \{V\}$ and give formulas for its curvature tensor, Ricci tensor and the scalar curvature. Next, we give some properties of the horizontally lifted vector fields and certain infinitesimal transformations. At the end, we consider some substructures of a $F(3, 1)$-structure on $\mathcal \{V\}$.},
author = {Anna Gasior},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Horizontal lift; $\pi $-conjugate connection; Killing field; infinitesimal transformation; $F(3, 1)$-structure; FK, FAK, FNK, FQK, FH-structure},
language = {eng},
number = {1},
pages = {null},
title = {Horizontal lift of symmetric connections to the bundle of volume forms $\mathcal \{V\}$},
url = {http://eudml.org/doc/289759},
volume = {54},
year = {2010},
}

TY - JOUR
AU - Anna Gasior
TI - Horizontal lift of symmetric connections to the bundle of volume forms $\mathcal {V}$
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2010
VL - 54
IS - 1
SP - null
AB - In this paper we present the horizontal lift of a symmetric affine connection with respect to another affine connection to the bundle of volume forms $\mathcal {V}$ and give formulas for its curvature tensor, Ricci tensor and the scalar curvature. Next, we give some properties of the horizontally lifted vector fields and certain infinitesimal transformations. At the end, we consider some substructures of a $F(3, 1)$-structure on $\mathcal {V}$.
LA - eng
KW - Horizontal lift; $\pi $-conjugate connection; Killing field; infinitesimal transformation; $F(3, 1)$-structure; FK, FAK, FNK, FQK, FH-structure
UR - http://eudml.org/doc/289759
ER -

References

top
  1. do Carmo, M. P., Riemannian Geometry, Graduate Texts in Mathematics 166, Birkhauser Boston Inc., Boston Ma., 1992. 
  2. Das, L. S., Complete lifts of a structure satisfying F K - ( - 1 ) K + 1 = 0 , Internat. J. Math. Math. Sci. 15 (1992), 803-808. 
  3. Dhooghe, P. F., The T. Y. Thomas construction of projectively related manifolds, Geom. Dedicata, 55 (1995), 221-235. 
  4. Dhooghe, P. F., Van Vlierden A., Projective geometry on the bundle of volume forms, J. Geom. 62 (1998), 66-83. 
  5. Gąsior, A., Curvatures for horizontal lift of Riemannian metric, Ann. Univ. Mariae Curie-Skłodowska Sect. A 60 (2006), 17-21. 
  6. Ishihara, S., Yano, K., Structure defined by f satisfying f 3 + f = 0 , Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) pp. 153-166 Nippon Hyoronsha, Tokyo, 1966. 
  7. Kobayashi. S., Nomizu, K., Foundations of Differential Geometry, John Wiley & Sons, New York-London, 1969. 
  8. Molino, P., Riemannian Foliations, Progression Mathematics, 73, Birkhauser Boston Inc., Boston Ma., 1988. 
  9. Miernowski A., Mozgawa W., Horizontal lift to the bundle of volume forms, Ann. Univ. Mariae Curie-Skłodowska Sect. A 57 (2003), 69-75. 
  10. Norden, A. P., Spaces with Affine Connection, Izdat. Nauka, Moscow, 1976 (Russian). 
  11. Radziszewski, K., π -geodesics and lines of shadow, Colloq. Math. 26 (1972), 157-163. 
  12. Rompała, W., Liftings of π -conjugate connections, Ann. Univ. Mariae Curie-Skłodowska Sect. A 32 (1978), 109-126. 
  13. Schouten, J. A., Ricci-Calculus, 2nd ed., Springer-Verlag, Berlin, Gottingen, Heidelberg, 1954. 
  14. Singh, K. D., Singh, R., Some f ( 3 , ε ) -structure manifolds, Demonstratio Math. 10 (1977), 637-645. 
  15. Yamauchi, K., On Riemannian manifolds admitting infinitesimal projective transformations, Hokkaido Math. J. 16 (1987), 115-125. 
  16. Yano, K., On structure defined by tensor field f of type (1, 1) satisfying f 3 + f = 0 , Tensor (N.S) 14 (1963), 99-109. 

NotesEmbed ?

top

You must be logged in to post comments.