Year: 2026
Type: article
Abstract: Let $$M^m$$ M m be a complete, connected, noncompact m -dimensional Riemannian manifold and $$\xi $$ ξ be a nontrivial closed conformal vector field on M , with at least one singular point, say p , and conformal factor $$\psi $$ ψ . We show that, when $$m>2$$ m > 2 or when $$m=2$$ m = 2 and the singular set of $$\xi $$ ξ consists of isolated points at which $$\psi $$ ψ does not vanish, then p is the only singular point of $$\xi $$ ξ and $$\exp _p:T_pM\rightarrow M$$ exp p : T p M → M is a diffeomorphism. Then, we use this fact to present a formula, built on $$|\xi |$$ | ξ | , for the Riemannian volume of geodesic balls of M centered at p . When $$m=2$$ m = 2 , such a formula generates necessary and sufficient conditions for M to be: (i) conformally equivalent to the Euclidean or hyperbolic plane; (ii) of finite total curvature. Finally, after showing that the conformal factor can be prescribed under some conditions, we finish the paper proving that $$\mathbb {C}^m$$ C m is the only example in the class of Kähler manifolds of complex dimension $$m>1$$ m > 1 .
Authors: Antonio Caminha, F. Yure Jansel
Institution: Universidade Federal do Ceará
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W7127896815