A sharp Eells-Sampson type theorem under positive sectional curvature upper bounds (Q6607347)

The main theorem of the authors is the following. \par Let \((M, g)\), \((N, h)\) be Riemannian manifolds, \((M, g)\) be closed (i.e., compact and without boundary), \(\varphi:M\rightarrow N\) a harmonic map. Assume that there exists \(K>0\) such that \(\mathrm{Ric}_g \ge(m-1)K\varphi^* h\) and \(\mathrm{sec}_h \le K\), where \(\mathrm{sec}_h\) denotes the various sectional curvatures of \(N\), and \(m=\dim M\). Then, \(\varphi\) is totally geodesic (i.e., carries geodesics of \(M\) to geodesics of \(N\)), and in particular, either\N\begin{itemize}\N\item[(i)] \(\varphi\) is constant, or\N\item[(ii)] \(\varphi\) is a homothetic immersion, \(g\) has positive constant curvature, \(\mathrm{Ric}_g =(m-1)K\varphi^* h\), and \(\mathrm{sec}_h (\prod)=K\) for every \(2\)-plane \(\prod\subseteq d\varphi(TM)\subseteq TN\).\N\end{itemize}\NThis result generalizes a theorem of \textit{J. Eells jun.} and \textit{J. H. Sampson} [Am. J. Math. 86, 109--160 (1964; Zbl 0122.40102)] for the case \(K=0\). \par In the context of harmonic-Einstein structures, this theorem has the following application (\(M\), \(N\), \(\varphi\) as above, \(m\ge 2\)). \par Let \((g, \varphi)\) be a harmonic-Einstein structure with respect to \(h\), i.e., \(\mathrm{Ric}_g -\alpha\varphi^* h=\lambda g\) for some \(\alpha\in{\mathbb R}\setminus\{0\}\), \(\lambda\in {\mathbb R}\). If \(\alpha >0\), \(\lambda\ge 0\), and \(\mathrm{sec}_h \le\frac{\alpha}{m-1}\), then \(\varphi\) is either constant or a homothetic immersion. In the latter case it holds that \(\lambda =0\), \(g\) has constant curvature, and \(\mathrm{sec}_h(\prod)=\frac{\alpha}{m-1}\) for every \(2\)-plane \(\prod\subseteq d\varphi(TM)\). \par Finally, as a consequence, the authors recover the following rigidity result of \textit{R. Hamilton} [in: Seminar on nonlinear partial differential equations, Publ., Math. Sci. Res. Inst. 2, 47--72 (1984; Zbl 0557.53018)]. \par Let \((M, h)\) be a closed Riemannian manifold with sectional curvature \(\le 1\). If there exists a Riemannian metric \(g\) on \(M\) such that \(\mathrm{Ric}_g =(m-1)h\), then the metrics \(g\) and \(h\) are homothetic (i.e., there exists a constant \(\mu >0\) such that \(g=\mu h\)) and the original metric \(h\) must have constant sectional curvature exactly \(1\) everywhere on \(M\). In particular, if \(\mathrm{sec}_h <1\) somewhere on \(M\), then there exists no metric \(g\) satisfying \(\mathrm{Ric}_g =(m-1)h\).