Harmonic maps as a subclass of isometric embeddings of the spacetime in five dimensions (Q702851)

The Campbell-Magaard theorem shows that any semi-Riemannian \(n\)-dimensional analytic manifold \((M^n, g)\) can be locally and isometrically embedded in a semi-Riemannian \((n+1)\)-dimensional analytic manifold \((N^{n+1}, h)\) whose Ricci curvature vanishes. In this paper, the authors considered minimality or harmonicity of such embedding. Introducing appropriate local coordinates on \(M\) and \(N\), respectively, and using the embedding \(\varphi = (\varphi^1, \dots, \varphi^{n+1})\) of \(M\) into \(N\), one can write \[ h = \bar{g}_{ij}(x, u) dx^i \otimes dx^j + du^2. \] Then a necessary and sufficient condition for an isometric embedding to be harmonic is that \[ \frac{1}{2} \partial_u \bar{g}_{ik} g^{ik} = 0\tag{1} \] The authors showed: a space-time \((M^4, g)\) can be harmonically embedded into a Ricci flat space-time \((N^5, h)\). They worked out an example to illustrate their result. More precisely, let \((M^4, g)\) be the Schwarzschild space-time with the Schwarzschild metric \(g\). Consider a five-dimensional Ricci flat space \((N^5 = M \times {\mathbb R}, h)\) with the metric given by \(ds_N^2 = f(u) g + du^2\), where \(f(u)\) is a differentiable function with \(f(0) = 1\) and \(f'(0) = 0\). Then this satisfies the equation (1) and so \((M^4, g)\) is harmonically and isometrically embedded into \((N^5, h)\) with \(u = 0\).