The Berger-Ebin theorem and harmonic maps and flows (Q1759269)

Let \(\left\langle M,g\right\rangle \) be a \(C^{\infty }\)-Riemannian manifold (\(\dim (M)\geq 2\)) and let \(\nabla \) denote the Levi-Civita connection on \(\left\langle M,g\right\rangle \). Then \(S^{p}M\) denotes the bundle of symmetric \(p\)-forms on \(M\) with the metric determined by \(g\), and \( C^{\infty }S^{p}M\) denotes the space of all \(C^{\infty }\)-sections of the bundle \(S^{p}M\). In local coordinates \(x^{1},x^{2},\dots,x^{n}\), the differential operator \(\delta ^{\ast }:C^{\infty }S^{p}M\rightarrow C^{\infty }S^{p+1}M\) is defined by \(\delta ^{\ast }\) \(\left( \varphi \right) _{i_{0}i_{1}\dots i_{p}}=\nabla _{i_{0}}\varphi _{i_{1}i_{2}\dots i_{p}}+\nabla _{i_{1}}\varphi _{i_{0}i_{2}\dots i_{p}}+\dots +\nabla _{i_{p}}\varphi _{i_{0}i_{1}i_{2}\dots i_{p-1}}\). Let \(\delta :C^{\infty }S^{p+1}M\rightarrow C^{\infty }S^{p}M\) denote the adjoint operator of \(\delta ^{\ast }\). The Berger-Ebin theorem [\textit{M. Berger} and \textit{D. Ebin}, J. Differ. Geom. 3, 379--392 (1969; Zbl 0194.53103)] states that \(C^{\infty }S^{2}M=\ker \delta \oplus \mathrm{im}\delta ^{\ast }\) on a closed oriented Riemannian manifold \(\left\langle M,g\right\rangle \), i.e., for every \(\varphi \in S^{2}M\), \(\varphi =\mathrm{proj}_{\ker \delta }\varphi +\mathrm{proj}_{\mathrm{im}\delta ^{\ast }}\varphi \); in addition, \( \mathrm{proj}_{\mathrm{im}\delta ^{\ast }}\varphi \) is given as the Lie derivative \(L_{X}g\) of the metric tensor \(g\) w.r.t. some vector field \(X\). Now we suppose that \(\left\langle M,g\right\rangle \) is a closed oriented \( C^{\infty }\)-Riemannian manifold with \(\dim \left( M\right) \geq 2\). A vector field \(X\in C^{\infty }TM\) is called an infinitesimal harmonic transformation of \(\left\langle M,g\right\rangle \) if the local one-parameter group of transformations generated by \(X\) consists of harmonic diffeomorphisms. Let \(f\in C^{\infty }\left( M,\overline{M}\right) \), where \( \left\langle \overline{M},\overline{g}\right\rangle \) is also a \(C^{\infty}\)-Riemannian manifold with Levi-Civita connection \(\overline{\nabla }\). Let \(f^{\ast }T\overline{M}\) be the bundle with base \(M\) and fibers \( T_{f\left( x\right) }\overline{M},x\in M\). Let \(\overline{g}^{\prime }\) be the standard metric on this bundle. Then \(g^{\ast }\) is defined as \(f^{\ast } \overline{g}\). The author applies the Berger-Ebin decomposition to \(g^{\ast } \) and the Ricci tensor \(\mathrm{Ric}\) to state and prove the following main theorems of his paper. Theorem 1. Let \(f:M\rightarrow \overline{M}\) be a smooth submersion or a diffeomorphism of a closed oriented Riemannian manifold \(\left\langle M,g\right\rangle \) onto a Riemannian manifold \(\left\langle \overline{M}, \overline{g}\right\rangle \) and let \(g^{\ast }=\mathrm{proj}_{\ker \delta }g^{\ast }+\mathrm{proj}_{\mathrm{im}\delta ^{\ast }}g^{\ast }\), where \(\mathrm{ proj}_{\mathrm{im}\delta ^{\ast }}g^{\ast }=L_{X}g\) for some \(X\in C^{\infty }TM\). Then if any two of the following three statements hold, then the third statement also holds: (i) \(f\) is harmonic; (ii) \(X\) is infinitesimal harmonic; (iii) \(\mathrm{div}X=\frac{1}{2}\mathrm{trace}_{g}\left( g^{\ast }\right) +\text{const}\). Theorem 2. Let \(\left\langle M,g\right\rangle \) be a closed oriented Riemannian manifold and let \(\mathrm{Ric}=\mathrm{proj}_{\ker \delta } \mathrm{Ric}+\mathrm{proj}_{\mathrm{im}\delta ^{\ast }}\mathrm{Ric}\), where \( \mathrm{proj}_{\mathrm{im}\delta ^{\ast }}\mathrm{Ric}=L_{Y}g\) for some \(Y\in C^{\infty }TM\). Then the following two statements \ hold: (i) \(Y\) is infinitesimal harmonic if and only if \(\mathrm{div}Y=\frac{1}{2}s+\text{const}\), where \(s\) is the scalar curvature of \(\left\langle M,g\right\rangle \); (ii) if \(L_{Y}s\leq 0\), then \(s=\text{const}\) and \(Y\) is a Killing vector field. In conclusion, two corollaries of Theorems 1 and 2 are given.