1.8 KiB
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.