Geometry of Harmonic Identity Maps

Year 2025, Volume: 18 Issue: 2, 396 - 402

Aicha Benkartab Ahmed Mohammed Cherif

https://doi.org/10.36890/iejg.1626396

Abstract

An identity map $(M,g)\longrightarrow(M,g)$ is a harmonic from a Riemannian manifold $(M,g)$ onto itself. In this paper, we study the harmonicity of identity maps $(M,g)\longrightarrow(M,g-df\otimes df)$ and $(M,g-df\otimes df)\longrightarrow(M,g)$ where $f$ is a smooth function with gradient norm $<1$ on $(M,g)$. We construct new examples of identity harmonic maps. We define a symmetric tensor field on $M$ whose properties are related to the harmonicity of these identity maps.

Keywords

Riemannian geometry , identity map , harmonic map

References

• Baird, P. and Wood,J. C. :Harmonic morphisms between Riemannain manifolds. Clarendon Press, Oxford (2003).
• Benkartab, A. and Mohammed Cherif, A.: New methods of construction for biharmonic maps, Kyungpook Math. J. 59(2019), 135–147.
• Benkartab, A. and Mohammed Cherif, A.: Deformations of Metrics and Biharmonic Maps, Commun. Math. 28 (2020), 263–275.
• Eells, J.and Lemaire, L.:A report on harmonic maps. Bull. London Math. Soc. 16, 1–68(1978).
• Eells, J.and Lemaire,L.: Another report on harmonic maps. Bull. London Math. Soc. 20 , 385–524(1988).
• Eells, J. and Sampson,J. H.: Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86, 109–160(1964).
• Merdji, B. and Mohammed Cherif, A.: New types of metrics deformations and applications to p-biharmonic maps. J. Indian Math. Soc. 90 (3-4) , 387–400(2023).
• Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications, Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994.
• Sakai, T.: Riemannian Geometry. Shokabo, Tokyo (1992). (in Japanese).
• Topping, P. : Lectures on the Ricci Flow. Number 325 in London Mathematical Society Lecture Note Series. Cambridge University Press, October, 2006.