\begin{definition}[Definition:Product Riemannian Metric]
Let $\struct {M_1, g_1}$ and $\struct {M_2, g_2}$ be Riemannian manifolds.
Let $p \in M$ be a point in $M$.
Let $T_p M$ be the tangent space of $M$ at $p$.
Let $v, w \in T_p M$ be tangent vectors.
Let $\tuple {v_1, v_2}, \tuple {w_1, w_2} \in T_{p_1} M_1 \oplus T_{p_2} M_2$.
Let $M_1 \times M_2$ be the product manifold.
Let $\tuple {p_1, p_2} \in M_1 \times M_2$ be a point in $M_1 \times M_2$.
Then the Riemannian metric $g = g_1 \oplus g_2$, called the '''product (Riemannian) metric''', is defined by
:$\map {g_{\tuple {p_1, p_2} } } {\tuple {v_1, v_2}, \tuple {w_1, w_2} } := \map {\valueat {g_1} {p_1} } {v_1, w_1} + \map {\valueat {g_2} {p_2} } {v_2, w_2}$
{{MissingLinks}}
{{NamedforDef|Bernhard Riemann|cat = Riemann}}
\end{definition}