Updated and added some to differential geometry.
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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
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## Curvature tensor
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> *Definition 2*: the Riemann curvature tensor $R: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}$ is defined as
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> *Definition 2*: the Riemann curvature tensor $\mathbf{R}: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}$ is defined as
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>
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> $$
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> R(\bm{\omega}, \mathbf{u}, \mathbf{v}, \mathbf{w}) = \mathbf{k}(\bm{\omega}, \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u}),
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> \mathbf{R}(\bm{\omega}, \mathbf{u}, \mathbf{v}, \mathbf{w}) = \mathbf{k}(\bm{\omega}, \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u}),
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> $$
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>
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> 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
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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.
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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 $\mathrm{TM}$.
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## Covariant derivative
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> *Definition 2*: let $\mathbf{v} = v^i \mathbf{e}_i\in \Gamma(\mathscr{B})$ then the **covariant derivative** on $\mathbf{v}$ is defined as
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@ -56,21 +58,88 @@ $$
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Will be added later.
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## Parallel transport
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## Intrinsic derivative
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> *Definition 3*: let $\mathbf{v} \in \Gamma(\mathrm{TM})$, then **parallel transport** of $\mathbf{v}$ occurs along the manifold $\mathrm{M}$ when
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> *Definition 3*: let $\gamma: \mathscr{D}(\gamma) \to M: t \mapsto \gamma(t)$ be a smooth curve on the manifold parameterized by an open interval $\mathscr{D}(\gamma) \subset \mathbb{R}$ and let $\mathbf{v}: \mathscr{D}(\gamma) \to \mathrm{TM}: t \mapsto \mathbf{v}(t) = \mathbf{u} \circ \gamma(t)$ be a vector field defined along the curve with $\mathbf{u} \in \Gamma(\mathrm{TM})$, the **intrinsic derivative** of $\mathbf{v}$ is defined as
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>
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> $$
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> D_k \mathbf{v} = \mathbf{0}.
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> D_t \mathbf{v}(t) = \nabla_{\dot\gamma} \mathbf{v}(t),
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> $$
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>
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> for all $t \in \mathscr{D}(\gamma)$.
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For example, a parameterised vector field $\mathbf{v}: x(t) \mapsto \mathbf{v}(x(t)) \in \Gamma(\mathrm{TM})$ is transported parallel if
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By decomposition of $\dot \gamma = \dot \gamma^i \partial_i$ and $\mathbf{v} = v^i \partial_i$ and using the chain rule we obtain
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$$
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D_t \mathbf{v} = (\partial_k v^i) \dot x^k \partial_i + \Gamma^i_{jk} v^j \partial_i = \mathbf{0},
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\begin{align*}
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\nabla_{\dot\gamma} \mathbf{v}(t) &= \dot \gamma^i \nabla_{\partial_i} (v^j \partial_j), \\
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&= \dot \gamma^i \big((\partial_i v^j) \partial_j + v^j \Gamma_{ji}^k \partial_k \big), \\
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&= (\dot \gamma^i \partial_i v^j + \dot \gamma^i \Gamma^j_{ki}v^k) \partial_j, \\
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&= (\dot v^j + \Gamma^j_{ki} v^k \dot \gamma^i) \partial_j,
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\end{align*}
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$$
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so $(\partial_k v^i) \dot x^k + \Gamma^i_{jk} v^j = 0$
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for all $t \in \mathscr{D}(\gamma)$. This notion of the intrinsic derivative can of course be extended to any tensor.
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### Parallel transport
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> *Definition 4*: let $\gamma: \mathscr{D}(\gamma) \to M: t \mapsto \gamma(t)$ be a smooth curve on the manifold parameterized by an open interval $\mathscr{D}(\gamma) \subset \mathbb{R}$ and let $\mathbf{v}: \mathbb{R} \to \mathrm{TM}: t \mapsto \mathbf{v}(t) = \mathbf{u} \circ \gamma(t)$ be a vector field defined along the curve with $\mathbf{u} \in \mathrm{TM}$, then **parallel transport** of $\mathbf{v}$ along the curve is defined as
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>
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> $$
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> D_t \mathbf{v}(t) = \mathbf{0},
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> $$
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>
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> for all $t \in \mathscr{D}(\gamma)$.
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Parallel transport implies the transport of a vector that is held constant along the path; constant direction and magnitude. It then follows that for $\dot \gamma = \dot \gamma^i \partial_i$ and $\mathbf{v} = v^i \partial_i$ parallel transport obtains
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$$
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D_t \mathbf{v}(t) = (\dot v^j + \Gamma^j_{ki} v^k \dot \gamma^i) \partial_j = \mathbf{0},
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$$
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obtaining the equations
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$$
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\dot v^j + \Gamma^j_{ki} v^k \dot \gamma^i = 0,
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$$
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such that
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$$
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\dot v^j = - \Gamma^j_{ki} v^k \dot \gamma^i,
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$$
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for all $t \in \mathscr{D}(\gamma)$. These equations can be solved for $\gamma$, obtaining the curve under which $\mathbf{v}$ stays constant.
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If we let $\mathbf{v} = \dot \gamma^i \partial_i$ be the tangent vector along the curve then parallel transport of $\mathbf{v}$ preserves the tangent vector and we obtain the **geodesic equations** given by
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$$
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\dot v^j + \Gamma^j_{ki} v^k \dot \gamma^i = \ddot\gamma^j + \Gamma^j_{ki} \dot\gamma^k \dot\gamma^i = 0,
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$$
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for all $t \in \mathscr{D}(\gamma)$.
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One may interpret a geodesic as a generalization of the notion of a straight line or shortest path defined by $\gamma$. As follows from the following proposition.
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> *Proposition 2*: let $\gamma: \mathscr{D}(\gamma) \to M: t \mapsto \gamma(t)$ be a smooth curve on the manifold parameterized by an open interval $\mathscr{D}(\gamma) \subset \mathbb{R}$ and let $\mathscr{L}$ be the Lagrangian defined by
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>
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> $$
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> \mathscr{L} = \|\dot \gamma\|^2 = g_{ij} \dot \gamma^i \dot \gamma^j,
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> $$
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>
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> for all $t \in \mathscr{D}(\gamma)$. By demanding [Hamilton's principle]() we obtain the geodesic equations
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>
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> $$
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> \ddot\gamma^j + \Gamma^j_{ki} \dot\gamma^k \dot\gamma^i = 0,
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> $$
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>
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> for all $t \in \mathscr{D}(\gamma)$.
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??? note "*Proof*:"
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Will be added later.
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It may be observed that by demanding the stationary state of the length of the curve we obtain the geodesic equations.
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## Contravariant derivative
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