Lengths and volumes

Let \(\mathrm{M}\) be a differential manifold with \(\dim \mathrm{M} = n \in \mathbb{N}\) used throughout the section. Let \(\mathrm{TM}\) and \(\mathrm{T^*M}\) denote the tangent and cotangent bundle, \(V\) and \(V^*\) the fiber and dual fiber bundle and \(\mathscr{B}\) the tensor fiber bundle.

Riemannian geometry

Definition 1: the length of a vector \(\mathbf{v} \in \Gamma(\mathrm{TM})\) is defined by the norm \(\|\cdot\|\) induced by the inner product \(\bm{g}\) such that

\[ \|\mathbf{v}\| = \sqrt{\bm{g}(\mathbf{v},\mathbf{v})}. \]

In the context of a smooth curve \(\mathbf{v}: \mathscr{D}(\mathbf{v}) \to \Gamma(\mathrm{TM}):t \mapsto \mathbf{v}(t)\) parameterized by an open interval \(\mathscr{D}(\mathbf{v}) \subset \mathbb{R}\), the length \(l_{12}\) of a closed section \([t_1, t_2] \subset \mathbb{R}\) of this curve is given by

\[ \begin{align*} l_{12} &= \int_{t_1}^{t_2} \|\mathbf{\dot v}(t)\| dt, \\ &= \int_{t_1}^{t_2} \sqrt{\bm{g}(\mathbf{\dot v},\mathbf{\dot v})} dt, \\ &= \int_{t_1}^{t_2} \sqrt{g_{ij} \dot v^i \dot v^j} dt, \end{align*} \]

with \(\mathbf{\dot v} = \dot v^i \partial_i \in \Gamma(\mathrm{TM})\).

Definition 2: the volume \(V\) span by the vectors \(\{\mathbf{v}_i\}_{i=1}^n\) in \(\Gamma(\mathrm{TM})\) is defined by

\[ V = \bm{\epsilon}(\mathbf{v}_1, \dots, \mathbf{v}_n) = \sqrt{g} \bm{\mu}(\mathbf{v}_1, \dots, \mathbf{v}_n), \]

with \(\bm{\epsilon}\) the unique unit volume form.

In the context of a subspace \(S \subset M\) with \(\dim S = k \in \mathbb{N}[k \leq n]\), the volume \(V\) is given by

\[ V = \int_S \bm{\epsilon} = \int_S \sqrt{g} dx^1 \dots dx^k. \]

It follows that for \(k=1\)

\[ \int_S \bm{\epsilon} = \int_S \sqrt{\bm{g}}. \]

Finsler geometry

Will be added later.