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Remarks on Scalar Curvature and Concircular Field Equation

Year 2021, , 121 - 124, 15.04.2021
https://doi.org/10.36890/iejg.906792

Abstract

We show that the scalar curvature of a Riemannian manifold $M$ is constant if it satisfies (i) the concircular field equation and $M$ is compact, (ii) the special concircular field equation. Finally, we show that, if a complete connected Riemannian manifold admits a concircular non-isometric vector field leaving the scalar curvature invariant, and the conformal function is special concircular, then the scalar curvature is a constant.

References

  • [1] Castro, I., Montealegre, C. R. and Urbano, F.: Closed conformal vector fields and Lagragian submanifolds. Pacific J. Math. Complex Space Forms. 199, 269-302 (2001).
  • [2] Chen, B.Y.: A simple characterization of generalized Robertason-Walker spacetimes. Gen. Rel. Grav. 46, 1833 (5 pp.) (2014).
  • [3] Chen, B.Y. and Deshmukh, S.: A note on Yamabe solitons. Balkan J. Geom. Applns. 23, 37-43 (2018).
  • [4] Daskalopoulos, P. and Sesum, N.: The classification of locally conformally flat Yamabe solitons. Adv. Math. 240, 346-369 (2013).
  • [5] Deshmukh, S. and Al-Solamy, F.: Conformal gradient vector fields on a compact Riemannian manifold. Colloq. Math. 112, 157-161 (2008).
  • [6] Fialkow, A.: Conformal geodesics. Trans. Amer. Math. Soc. 45, 443-473 (1989).
  • [7] Hamilton, R.S.: The Ricci flow on surfaces, Mathematics and general relativity. Contemp. Math. 71, 237-262 (1988).
  • [8] Obata, M.: Conformal transformations of Riemannian manifolds. J. Differential Geom. 4, 311-333 (1970).
  • [9] Ros, A. and Urbano, F.: Lagrangian submanifolds of Cn with confornal Maslov form and the Whitney sphere. J. Math. Soc. Japan. 50, 203-226 (1998).
  • [10] Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117, 251-275 (1965).
Year 2021, , 121 - 124, 15.04.2021
https://doi.org/10.36890/iejg.906792

Abstract

References

  • [1] Castro, I., Montealegre, C. R. and Urbano, F.: Closed conformal vector fields and Lagragian submanifolds. Pacific J. Math. Complex Space Forms. 199, 269-302 (2001).
  • [2] Chen, B.Y.: A simple characterization of generalized Robertason-Walker spacetimes. Gen. Rel. Grav. 46, 1833 (5 pp.) (2014).
  • [3] Chen, B.Y. and Deshmukh, S.: A note on Yamabe solitons. Balkan J. Geom. Applns. 23, 37-43 (2018).
  • [4] Daskalopoulos, P. and Sesum, N.: The classification of locally conformally flat Yamabe solitons. Adv. Math. 240, 346-369 (2013).
  • [5] Deshmukh, S. and Al-Solamy, F.: Conformal gradient vector fields on a compact Riemannian manifold. Colloq. Math. 112, 157-161 (2008).
  • [6] Fialkow, A.: Conformal geodesics. Trans. Amer. Math. Soc. 45, 443-473 (1989).
  • [7] Hamilton, R.S.: The Ricci flow on surfaces, Mathematics and general relativity. Contemp. Math. 71, 237-262 (1988).
  • [8] Obata, M.: Conformal transformations of Riemannian manifolds. J. Differential Geom. 4, 311-333 (1970).
  • [9] Ros, A. and Urbano, F.: Lagrangian submanifolds of Cn with confornal Maslov form and the Whitney sphere. J. Math. Soc. Japan. 50, 203-226 (1998).
  • [10] Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117, 251-275 (1965).
There are 10 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Ramesh Sharma This is me 0000-0003-3700-8164

Sharief Deshmukh

Publication Date April 15, 2021
Acceptance Date December 26, 2020
Published in Issue Year 2021

Cite

APA Sharma, R., & Deshmukh, S. (2021). Remarks on Scalar Curvature and Concircular Field Equation. International Electronic Journal of Geometry, 14(1), 121-124. https://doi.org/10.36890/iejg.906792
AMA Sharma R, Deshmukh S. Remarks on Scalar Curvature and Concircular Field Equation. Int. Electron. J. Geom. April 2021;14(1):121-124. doi:10.36890/iejg.906792
Chicago Sharma, Ramesh, and Sharief Deshmukh. “Remarks on Scalar Curvature and Concircular Field Equation”. International Electronic Journal of Geometry 14, no. 1 (April 2021): 121-24. https://doi.org/10.36890/iejg.906792.
EndNote Sharma R, Deshmukh S (April 1, 2021) Remarks on Scalar Curvature and Concircular Field Equation. International Electronic Journal of Geometry 14 1 121–124.
IEEE R. Sharma and S. Deshmukh, “Remarks on Scalar Curvature and Concircular Field Equation”, Int. Electron. J. Geom., vol. 14, no. 1, pp. 121–124, 2021, doi: 10.36890/iejg.906792.
ISNAD Sharma, Ramesh - Deshmukh, Sharief. “Remarks on Scalar Curvature and Concircular Field Equation”. International Electronic Journal of Geometry 14/1 (April 2021), 121-124. https://doi.org/10.36890/iejg.906792.
JAMA Sharma R, Deshmukh S. Remarks on Scalar Curvature and Concircular Field Equation. Int. Electron. J. Geom. 2021;14:121–124.
MLA Sharma, Ramesh and Sharief Deshmukh. “Remarks on Scalar Curvature and Concircular Field Equation”. International Electronic Journal of Geometry, vol. 14, no. 1, 2021, pp. 121-4, doi:10.36890/iejg.906792.
Vancouver Sharma R, Deshmukh S. Remarks on Scalar Curvature and Concircular Field Equation. Int. Electron. J. Geom. 2021;14(1):121-4.