Geodesic
From infinitesimal harmonic transformations to Ricci solitons
Mathematica Bohemica, Tome 138 (2013) no. 1, pp. 25-36

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
The concept of the Ricci soliton was introduced by R. S. Hamilton. The Ricci soliton is defined by a vector field and it is a natural generalization of the Einstein metric. We have shown earlier that the vector field of the Ricci soliton is an infinitesimal harmonic transformation. In our paper, we survey Ricci solitons geometry as an application of the theory of infinitesimal harmonic transformations.
The concept of the Ricci soliton was introduced by R. S. Hamilton. The Ricci soliton is defined by a vector field and it is a natural generalization of the Einstein metric. We have shown earlier that the vector field of the Ricci soliton is an infinitesimal harmonic transformation. In our paper, we survey Ricci solitons geometry as an application of the theory of infinitesimal harmonic transformations.
DOI : 10.21136/MB.2013.143227
Classification : 53C20, 53C25, 53C43
Keywords: Ricci soliton; infinitesimal harmonic transformation; Riemannian manifold 
Stepanov, Sergey E.; Tsyganok, Irina I.; Mikeš, Josef. From infinitesimal harmonic transformations to Ricci solitons. Mathematica Bohemica, Tome 138 (2013) no. 1, pp. 25-36. doi: 10.21136/MB.2013.143227
@article{10_21136_MB_2013_143227,
     author = {Stepanov, Sergey E. and Tsyganok, Irina I. and Mike\v{s}, Josef},
     title = {From infinitesimal harmonic transformations to {Ricci} solitons},
     journal = {Mathematica Bohemica},
     pages = {25--36},
     year = {2013},
     volume = {138},
     number = {1},
     doi = {10.21136/MB.2013.143227},
     mrnumber = {3076218},
     zbl = {1274.53096},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2013.143227/}
}
TY  - JOUR
AU  - Stepanov, Sergey E.
AU  - Tsyganok, Irina I.
AU  - Mikeš, Josef
TI  - From infinitesimal harmonic transformations to Ricci solitons
JO  - Mathematica Bohemica
PY  - 2013
SP  - 25
EP  - 36
VL  - 138
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2013.143227/
DO  - 10.21136/MB.2013.143227
LA  - en
ID  - 10_21136_MB_2013_143227
ER  - 
%0 Journal Article
%A Stepanov, Sergey E.
%A Tsyganok, Irina I.
%A Mikeš, Josef
%T From infinitesimal harmonic transformations to Ricci solitons
%J Mathematica Bohemica
%D 2013
%P 25-36
%V 138
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2013.143227/
%R 10.21136/MB.2013.143227
%G en
%F 10_21136_MB_2013_143227

[1] Bochner, S.: Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52 (1946), 776-797. | DOI | MR | Zbl

[2] Chow, B., Knopf, D.: The Ricci Flow: an Introduction. Mathematical Surveys and Monographs 110, American Mathematical Society, Providence, RI (2004), 325. | MR | Zbl

[3] Chow, B., Lu, P., Ni, L.: Hamilton's Ricci Flow. AMS Bookstore (2006), 608. | MR | Zbl

[4] Eells, J., Lemaire, L.: A report on harmonic maps. Bull. London Math. Soc. 10 (1978), 1-68. | DOI | MR | Zbl

[5] Ezin, J. P., Bourguignon, J. P.: Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. Amer. Math. Soc. 301 (1987), 723-736. | DOI | MR | Zbl

[6] Eminent, M., Nave, G. La, Mantegazza, C.: Ricci solitons---the equation point of view. Manuscript Math. 127 (2008), 345-367. | DOI | MR

[7] Gray, A.: Nearly Kähler manifolds. J. Differ. Geom. 4 (1970), 283-309. | DOI | MR | Zbl

[8] Hamilton, R. S.: The Ricci flow on surface. Mathematics and general relativity (Proc. Conf. Santa Cruz/Calif., 1986), Contemp. Math. 71 (1988), 237-262. | DOI | MR

[9] Hamilton, R. S.: The formation of singularities in the Ricci flow. (Cambridge, MA, USA, 1993). Suppl. J. Differ. Geom. 2 (1995), 7-136. | MR | Zbl

[10] Hsiung, C.: On the group of conformal transformations of a compact Riemannian manifold. Proc. Natl. Acad. Sci. USA 54 (1965), 1509-1513. | DOI | MR | Zbl

[11] Ivey, T.: Ricci solitons on compact three-manifolds. Diff. Geom. Appl. 3 (1993), 301-307. | DOI | MR | Zbl

[12] Ishihara, S., Tashiro, Y.: On Riemannian manifolds admitting a concircular transformation. Math. J. Okayama Univ. 9 (1959), 19-47. | MR | Zbl

[13] Kobayashi, K.: Transformation Group in Differential Geometry. Springer, Berlin (1972), 182. | MR

[14] Lichnerowicz, A.: Sur les tranformations conformes d'une variété riemannianne compacte. French C.R. Acad. Sci. Paris 259 (1964), 697-700. | MR

[15] Nouhaud, O.: Transformations infinitesimales harmoniques. C. R. Acad., Paris, Ser. A 274 (1972), 573-576. | MR | Zbl

[16] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159v1 [math.DG] 11 Nov 2002 39. | Zbl

[17] Petersen, D.: Riemannian Geometry. 2nd ed. Springer, New York (2006), 401. | MR | Zbl

[18] Smol'nikova, M. V.: On global geometry of harmonic symmetric bilinear forms. Proc. Steklov Inst. Math. 236 (2002), 315-318. | MR

[19] Stepanov, S. E., Smol'nikova, M. V., Shandra, I. G.: Infinitesimal harmonic maps. Russ. Math. 48 (2004), 65-70. | MR | Zbl

[20] Stepanov, S. E., Shandra, I. G.: Geometry of infinitesimal harmonic transformations. Ann. Global Anal. Geom. 24 (2003), 291-299. | DOI | MR | Zbl

[21] Stepanov, S. E., Shelepova, V. N.: A note on Ricci soliton. Mathematical Notes 86 (2009), 447-450. | DOI | MR

[22] Yano, K.: The Theory of Lie Derivatives and Its Applications. Nord-Holland, Amsterdam (1957), 299. | MR | Zbl

[23] Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970), 156. | MR | Zbl

[24] Yano, K., Nagano, T.: On geodesic vector fields in a compact orientable Riemannian space. Comment. Math. Helv. 35 (1961), 55-64. | DOI | MR

[25] Yano, K.: Differential Geometry on Complex and Almost Complex Spaces. Pergamon Press, Oxford (1965), 323. | MR | Zbl

[26] Yau, S.-T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25 (1976), 659-670. | DOI | MR | Zbl

[27] Zhang, Z.-H.: Gradient shrinking solitons with vanishing Weyl tensor. Pac. J. Math. 242 (2009), 189-200. | DOI | MR | Zbl

Cité par Sources :