Finding a Riemannian metric in the case when some of the components of a curvature's tensor don't depend on one of the variables (Q2712490)

The existence and uniqueness of Riemannian metrics \(g_{ij} (x)\) \((x\in D\subset \mathbb{R}^n;i,j=1, \dots,n)\) are investigated if the values \(g_{ij} |_{x^1=0}, {\partial g_{ij}\over \partial x^1}|_{x^1=0}\), \(g_{ij} |_{x^r=0}\) \((i,j=2, \dots,n\); \(r\) fixed, \(2\leq r\leq n)\) are given and some components of the curvature tensors \(R^i_{jkl}\) are assumed independent of \(x^r\). (In more detail. Let \(I\) be the set of all pairs \((k,m)\) with \(2\leq k\), \(m\leq n\). Let \(I'\subset I\) be a subset, \(I''=I-I'\) the complementary subset. We denote \(b_{ij}=R^1_{i1j}\) (if \((i,j)\in I')\), \(b_{ij}= R^1_{ij1}\) (if \((i,j)\in I'')\), \(c_{ij}= R^1_{ij1}\) or \(R^1_{i1j}\) (if \((i,j)\in I'')\), \(c_{ij}= R^1_{1ij}\) (if \((i,j)\in I')\). It is assumed that at least one of the vector functions \((b_{ij})\) or \((c_{ij})\) does not depend on \(x^r\).) Rather intricate adaptations of the relevant equations \(R^i_{qks}= \partial \Gamma^i_{q k}/ \partial x^s- \partial\Gamma^i_{qs}/\partial x^k+ \Gamma^q_{qk}\Gamma^i_{ps} -\Gamma^p_{qs} \Gamma^p_{pq}\) are employed.