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Updated and added some to differential geometry.

This commit is contained in:
Luc Bijl 2024-05-25 13:26:37 +02:00
parent ba283597d6
commit cf378f5791
2 changed files with 77 additions and 8 deletions

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@ -28,10 +28,10 @@ It then follows from the definition that the curvature operator $\Omega$ can be
## Curvature tensor
> *Definition 2*: the Riemann curvature tensor $R: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}$ is defined as
> *Definition 2*: the Riemann curvature tensor $\mathbf{R}: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}$ is defined as
>
> $$
> R(\bm{\omega}, \mathbf{u}, \mathbf{v}, \mathbf{w}) = \mathbf{k}(\bm{\omega}, \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u}),
> \mathbf{R}(\bm{\omega}, \mathbf{u}, \mathbf{v}, \mathbf{w}) = \mathbf{k}(\bm{\omega}, \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u}),
> $$
>
> for all $\bm{\omega} \in \Gamma(\mathrm{T}^*\mathrm{M})$ and $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM})$.

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@ -16,6 +16,8 @@ Let $\mathrm{M}$ be a differential manifold with $\dim \mathrm{M} = n \in \mathb
From property 3 it becomes clear that $\nabla_\mathbf{v}$ is an analogue of a directional derivative. The linear connection can also be defined in terms of the cotangent bundle and the dual fiber bundle.
Note that the first (trivial) element in the notion of the section $\Gamma$ is omitted, generally it should be $\Gamma(\mathrm{M}, \mathrm{TM})$ as the elements of this set are maps from $\mathrm{M}$ to $\mathrm{TM}$.
## Covariant derivative
> *Definition 2*: let $\mathbf{v} = v^i \mathbf{e}_i\in \Gamma(\mathscr{B})$ then the **covariant derivative** on $\mathbf{v}$ is defined as
@ -56,21 +58,88 @@ $$
Will be added later.
## Parallel transport
## Intrinsic derivative
> *Definition 3*: let $\mathbf{v} \in \Gamma(\mathrm{TM})$, then **parallel transport** of $\mathbf{v}$ occurs along the manifold $\mathrm{M}$ when
> *Definition 3*: let $\gamma: \mathscr{D}(\gamma) \to M: t \mapsto \gamma(t)$ be a smooth curve on the manifold parameterized by an open interval $\mathscr{D}(\gamma) \subset \mathbb{R}$ and let $\mathbf{v}: \mathscr{D}(\gamma) \to \mathrm{TM}: t \mapsto \mathbf{v}(t) = \mathbf{u} \circ \gamma(t)$ be a vector field defined along the curve with $\mathbf{u} \in \Gamma(\mathrm{TM})$, the **intrinsic derivative** of $\mathbf{v}$ is defined as
>
> $$
> D_k \mathbf{v} = \mathbf{0}.
> D_t \mathbf{v}(t) = \nabla_{\dot\gamma} \mathbf{v}(t),
> $$
>
> for all $t \in \mathscr{D}(\gamma)$.
For example, a parameterised vector field $\mathbf{v}: x(t) \mapsto \mathbf{v}(x(t)) \in \Gamma(\mathrm{TM})$ is transported parallel if
By decomposition of $\dot \gamma = \dot \gamma^i \partial_i$ and $\mathbf{v} = v^i \partial_i$ and using the chain rule we obtain
$$
D_t \mathbf{v} = (\partial_k v^i) \dot x^k \partial_i + \Gamma^i_{jk} v^j \partial_i = \mathbf{0},
\begin{align*}
\nabla_{\dot\gamma} \mathbf{v}(t) &= \dot \gamma^i \nabla_{\partial_i} (v^j \partial_j), \\
&= \dot \gamma^i \big((\partial_i v^j) \partial_j + v^j \Gamma_{ji}^k \partial_k \big), \\
&= (\dot \gamma^i \partial_i v^j + \dot \gamma^i \Gamma^j_{ki}v^k) \partial_j, \\
&= (\dot v^j + \Gamma^j_{ki} v^k \dot \gamma^i) \partial_j,
\end{align*}
$$
so $(\partial_k v^i) \dot x^k + \Gamma^i_{jk} v^j = 0$
for all $t \in \mathscr{D}(\gamma)$. This notion of the intrinsic derivative can of course be extended to any tensor.
### Parallel transport
> *Definition 4*: let $\gamma: \mathscr{D}(\gamma) \to M: t \mapsto \gamma(t)$ be a smooth curve on the manifold parameterized by an open interval $\mathscr{D}(\gamma) \subset \mathbb{R}$ and let $\mathbf{v}: \mathbb{R} \to \mathrm{TM}: t \mapsto \mathbf{v}(t) = \mathbf{u} \circ \gamma(t)$ be a vector field defined along the curve with $\mathbf{u} \in \mathrm{TM}$, then **parallel transport** of $\mathbf{v}$ along the curve is defined as
>
> $$
> D_t \mathbf{v}(t) = \mathbf{0},
> $$
>
> for all $t \in \mathscr{D}(\gamma)$.
Parallel transport implies the transport of a vector that is held constant along the path; constant direction and magnitude. It then follows that for $\dot \gamma = \dot \gamma^i \partial_i$ and $\mathbf{v} = v^i \partial_i$ parallel transport obtains
$$
D_t \mathbf{v}(t) = (\dot v^j + \Gamma^j_{ki} v^k \dot \gamma^i) \partial_j = \mathbf{0},
$$
obtaining the equations
$$
\dot v^j + \Gamma^j_{ki} v^k \dot \gamma^i = 0,
$$
such that
$$
\dot v^j = - \Gamma^j_{ki} v^k \dot \gamma^i,
$$
for all $t \in \mathscr{D}(\gamma)$. These equations can be solved for $\gamma$, obtaining the curve under which $\mathbf{v}$ stays constant.
If we let $\mathbf{v} = \dot \gamma^i \partial_i$ be the tangent vector along the curve then parallel transport of $\mathbf{v}$ preserves the tangent vector and we obtain the **geodesic equations** given by
$$
\dot v^j + \Gamma^j_{ki} v^k \dot \gamma^i = \ddot\gamma^j + \Gamma^j_{ki} \dot\gamma^k \dot\gamma^i = 0,
$$
for all $t \in \mathscr{D}(\gamma)$.
One may interpret a geodesic as a generalization of the notion of a straight line or shortest path defined by $\gamma$. As follows from the following proposition.
> *Proposition 2*: let $\gamma: \mathscr{D}(\gamma) \to M: t \mapsto \gamma(t)$ be a smooth curve on the manifold parameterized by an open interval $\mathscr{D}(\gamma) \subset \mathbb{R}$ and let $\mathscr{L}$ be the Lagrangian defined by
>
> $$
> \mathscr{L} = \|\dot \gamma\|^2 = g_{ij} \dot \gamma^i \dot \gamma^j,
> $$
>
> for all $t \in \mathscr{D}(\gamma)$. By demanding [Hamilton's principle]() we obtain the geodesic equations
>
> $$
> \ddot\gamma^j + \Gamma^j_{ki} \dot\gamma^k \dot\gamma^i = 0,
> $$
>
> for all $t \in \mathscr{D}(\gamma)$.
??? note "*Proof*:"
Will be added later.
It may be observed that by demanding the stationary state of the length of the curve we obtain the geodesic equations.
## Contravariant derivative