notes/docs/mathematics/differential-geometry/lengths-and-volumes.md

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.