## Journal of Differential Geometry

### The harmonic field of a Riemannian manifold

Steve Halperin

#### Abstract

Sullivan’s construction of minimal models for topological spaces is refined for the case of a simply connected closed Riemannian manifold, $(M,\lt,\gt)$, to define a unique finitely generated field extension, $\mathbf{k}$ of $\mathbf{Q}$, baptized the harmonic field of $(M,\lt,\gt)$, and a morphism, $m:(\Lambda V,d)\rightarrow A_{DR}(M)$, from a Sullivan model defined over $\mathbf{k}$. The Sullivan model and the morphism are determined up to isomorphism, and the natural extension of $H(m)$ to $H(\Lambda V,d)\otimes_{\mathbf{k}}\mathbf{R}$ is an isomorphism; in particular, $(\Lambda V,d)$ is isomorphic to a rational Sullivan model for $M$ tensored with $\mathbf{k}$. Examples are constructed to show that every finitely generated extension field of $\mathbf{Q}$ occurs as a harmonic field of such a Riemannian manifold.

#### Article information

Source
J. Differential Geom., Volume 96, Number 1 (2014), 61-76.

Dates
First available in Project Euclid: 31 January 2014

https://projecteuclid.org/euclid.jdg/1391192692

Digital Object Identifier
doi:10.4310/jdg/1391192692

Mathematical Reviews number (MathSciNet)
MR3161385

Zentralblatt MATH identifier
1310.53032

#### Citation

Halperin, Steve. The harmonic field of a Riemannian manifold. J. Differential Geom. 96 (2014), no. 1, 61--76. doi:10.4310/jdg/1391192692. https://projecteuclid.org/euclid.jdg/1391192692