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Let $\\mathrm{T}^1(V_i)$ denote the unit tangent bundle of $V_i$ for each $i=1,\\dots , k$, then for every $(v_1, \\dots , v_k) \\in \\mathrm{T}^1 (V_1) \\times \\cdots \\times \\mathrm{T}^1 (V_k)$, the diagonal geodesic flow $g_t$ is defined by $g_t (v_1, \\dots , v_k) = (g_t v_1, \\dots , g_t v_k)$. 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