Definition:Warped Product Manifold

Definition

Let $\struct {M_1, g_1}$, $\struct {M_2, g_2}$ be Riemannian manifolds.

Let $f : M_1 \to \R_{\mathop > 0}$ be a strictly positive smooth function.

Let $p \in M$ be a point.

Let $T_p M$ be a tangent space of $M$ at $p$.

Let $v \in T_p M$ be a tangent vector.

Let $g_p$ be a Riemannian metric of $M$ at $p$.

Then the warped product manifold $M_1 \times_f M_2$ is a product manifold $M_1 \times M_2$ endowed with the Riemannian metric $g = g_1 \oplus f^2 g_2$ such that:

$\forall p_1 \in M_1 : \forall p_2 \in M_2 : \forall \tuple {v_1, v_2}, \tuple {w_1, w_2} \in T_{p_1} M_1 \oplus T_{p_2} M_2 : \map {g_{\tuple {p_1, p_2} } } {\tuple {v_1, v_2}, \tuple {w_1, w_2}} = \map {\bigvalueat {g_1} {p_1}} {v_1, w_1} + \map {f^2} {p_1} \map {\bigvalueat {g_2} {p_2}} {v_2, w_2}$

where $\times$ is the Cartesian product and $\oplus$ is the direct sum.

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Sources

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