On Riemannian submersions (Q2322498)

If \((X, h)\) and \((Y, k)\) are Riemannian manifolds, then a \(C^1\) map \(\varphi:X\to Y\) is called a Riemannian submersion if for every \(x\in X\) the differential \(\varphi_*\) induces the metric on \(T_{\varphi(x)}Y\) from the metric of \(T_xX\). In other words, \(\varphi_*\) maps the orthogonal complement of \(\text{ker}\varphi_*\) isometrically on \(TY\).\par In this paper, the author considers Riemannian submersions between Riemannian manifolds. First, it is shown that if \((X,h)\) is a real analytic and \((Y,k)\) is a \(C^\infty\) Riemannian manifold, both finite dimensional, and \(\varphi:X\to Y\) is a surjective \(C^\infty\) Riemannian submersion, then \(Y\) admits a real analytic structure in which \(k\) is real analytic.\par Next, the author deals with the submersion \(\varphi:X\to Y\) if \((Y,k)\) is only a \(C^p\) Riemannian manifold, and he proves that if \(p\in(3,\infty]\) is nonintegral, \(Y\) is a \(C^{p+1}\) manifold endowed with a \(C^p\) Riemannian metric \(k\), \((X,h)\) is an analytic Riemannian manifold, and \(\varphi:X\to Y\) is a surjective Riemannian submersion of class \(C^{p+1}\), then \(Y\) admits an analytic structure in which \(h\) is analytic. Moreover, this analytic structure and the original \(C^{p+1}\) manifold structure of \(Y\) share the same underlying \(C^p\) manifold structure.\par The proof of this property uses so-called adapted complex structures, that is, Kähler structures on the tangent of a cotangent bundle of a Riemannian manifold.