Definition:Product Riemannian Metric

Definition

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

$\map {g_{\tuple {p_1, p_2} } } {\tuple {v_1, v_2}, \tuple {w_1, w_2} } := \map {\valueat {g_1} {p_1} } {v_1, w_1} + \map {\valueat {g_2} {p_2} } {v_2, w_2}$

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Source of Name

This entry was named for Bernhard Riemann.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics