Geodesics of a Riemannian product

Choose product coordinates $(x^1,\dots ,x^m,y^1,\dots ,y^n)$ and write $\bar{G}$ for the full product-metric matrix. By [F1], the metric and its inverse have matrices $\bar{G}=\left(\begin{array}{cc}(g_{ij}(x))& 0\\ 0& (h_{\alpha \beta }(y))\end{array}\right),{\bar{G}}^{-1}=\left(\begin{array}{cc}(g^{ij}(x))& 0\\ 0& (h^{\alpha \beta }(y))\end{array}\right)\mathrm{.}$ In particular, $g_{ij}$ has no $y$-dependence, $h_{\alpha \beta }$ has no $x$-dependence, and every mixed metric coefficient is zero.

By [F2], the Christoffel formula applied to the first block gives ${\mathrm{\Gamma }}^k{}_{ij}={\mathrm{\Gamma }}^k{}_{ij}(g)$, and its application to the second gives ${\mathrm{\Gamma }}^{\gamma }{}_{\alpha \beta }={\mathrm{\Gamma }}^{\gamma }{}_{\alpha \beta }(h)$. Every symbol whose indices meet both blocks vanishes. For example, ${\mathrm{\Gamma }}^k{}_{i\beta }=\frac{1}{2}{g}^{k\mathrm{\ell }}({\mathrm{\partial }}_i{\bar{G}}_{\beta \mathrm{\ell }}+{\mathrm{\partial }}_{\beta }{g}_{i\mathrm{\ell }}-{\mathrm{\partial }}_{\mathrm{\ell }}{\bar{G}}_{i\beta })=0,$ and ${\mathrm{\Gamma }}^k{}_{\alpha \beta }=-\frac{1}{2}{g}^{k\mathrm{\ell }}{\mathrm{\partial }}_{\mathrm{\ell }}{h}_{\alpha \beta }=0;$ the cases with an upper $N$-index are identical with the two factors exchanged. Thus the product Levi--Civita symbols are precisely the two factor families, with zero mixed symbols.

Write the coordinate functions of $\alpha$ and $\beta$ as $x^i(t)$ and $y^{\alpha }(t)$. Substituting step 2.1 into [F3], the product geodesic equations split into the two independent systems ${\ddot{x}}^k+{\mathrm{\Gamma }}^k{}_{ij}(g)(x){\dot{x}}^i{\dot{x}}^j=0(1\le k\le m),{\ddot{y}}^{\gamma }+{\mathrm{\Gamma }}^{\gamma }{}_{\alpha \beta }(h)(y){\dot{y}}^{\alpha }{\dot{y}}^{\beta }=0(1\le \gamma \le n)\mathrm{.}$ These are exactly the coordinate geodesic equations for $\alpha$ and $\beta$, evaluated at the same value of $t$.

If $\gamma$ is a geodesic, [F3] and step 3.1 make both factor systems vanish, so $\alpha$ and $\beta$ are geodesics with the same affine parameter.

Conversely, if both factor curves are geodesics in the supplied parameter, both systems in step 3.1 vanish on every product-chart subinterval, and [F3] makes $\gamma$ a product geodesic. Taking $I=\mathbb{R}$ in steps 4.1--4.2 proves the all-real assertion in both directions.

Under the additional hypotheses stated there, [A1] and [F4] applied to the two-factor family give the completeness consequence. The geodesic iff in steps 1.1--4.2 itself uses no choice: all coordinates and the curve are supplied, and the Levi--Civita connection is uniquely determined. If either factor is empty, there is no supplied curve from a nonempty interval and the universal iff is vacuous; the conditional completeness clause explicitly excludes that case. A zero-dimensional factor contributes an empty coordinate system and a locally constant component, while a one-dimensional factor contributes its single geodesic equation. Constant components, including two constant components, satisfy their factor equations, so all degenerate cases are included. Included endpoints use the one-sided convention in [F3]. Steps 4.1 and 4.2 respectively establish the forward and reverse implications, and neither direction changes or independently rescales the common parameter.