Bounds of Riemannian Metric by Euclidean Metric on Euclidean Space

Theorem

Let $W \subseteq \R^n$ be a subset of an $n$-dimensional Riemannian manifold.

Let $g$ be a Riemannian metric on $W$.

Let $\tilde g$ be a Euclidean metric on $W$.

Let $K \subseteq W$ be a compact subset.

Let $c, C \in \R_{\mathop > 0}$ be positive constants.

Let $T_x \R^n$ be the tangent space of $\R^n$ at $x \in \R^n$.

Let $\size {\, \cdot \,}_g$ be the Riemannian inner product norm.

Then:

$\forall K \subseteq W : \exists c, C \in \R_{\mathop > 0} : \forall x \in K : \forall v \in T_x \R^n : c \size v_{\tilde g} \le \size v_g \le C \size v_{\tilde g}$

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Lengths and Distances