From ba283597d688979f16cfffb0ff4206be00f9ca39 Mon Sep 17 00:00:00 2001 From: Luc Date: Thu, 23 May 2024 20:42:45 +0200 Subject: [PATCH] Partly finished differential geometry. --- config/en/mkdocs.yaml | 4 +- .../differential-geometry/curvature.md | 38 +++++++++++ .../differential-geometry/derivatives.md | 60 +++++++++++++++++ .../linear-connections.md | 65 ++++++++++++++++++- .../differential-geometry/torsion.md | 39 +++++++++++ 5 files changed, 201 insertions(+), 5 deletions(-) diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index ce895dd..cbf3c14 100755 --- a/config/en/mkdocs.yaml +++ b/config/en/mkdocs.yaml @@ -97,8 +97,8 @@ nav: - 'Tensor symmetries': mathematics/linear-algebra/tensors/tensor-symmetries.md - 'Volume forms': mathematics/linear-algebra/tensors/volume-forms.md - 'Tensor transformations': mathematics/linear-algebra/tensors/tensor-transformations.md - - 'Topology': - - 'Fiber bundles': mathematics/topology/fiber-bundles.md + - 'Topology': + - 'Fiber bundles': mathematics/topology/fiber-bundles.md - 'Calculus': - 'Limits': mathematics/calculus/limits.md - 'Continuity': mathematics/calculus/continuity.md diff --git a/docs/en/mathematics/differential-geometry/curvature.md b/docs/en/mathematics/differential-geometry/curvature.md index e69de29..02fb4c0 100644 --- a/docs/en/mathematics/differential-geometry/curvature.md +++ b/docs/en/mathematics/differential-geometry/curvature.md @@ -0,0 +1,38 @@ +# Curvature + +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. + +## Curvature operator + +> *Definition 1*: the **curvature operator** $\Omega: \Gamma(\mathrm{TM})^3 \to \Gamma(\mathrm{TM})$ is defined as +> +> $$ +> \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u} = [\nabla_\mathbf{v}, \nabla_\mathbf{w}] \mathbf{u} - \nabla_{[\mathbf{v}, \mathbf{w}]}\mathbf{u}, +> $$ +> +> for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM})$ with $[\cdot, \cdot]$ denoting the [Lie bracket](). + +It then follows from the definition that the curvature operator $\Omega$ can be decomposed. + +> *Proposition 1*: the decomposition of the curvature operator $\Omega$ relative to a basis $\{\partial_i\}_{i=1}^n$ of $\Gamma(\mathrm{TM})$ results into +> +> $$ +> \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u} = v^i w^j [D_i, D_j] u^l \partial_l, +> $$ +> +> for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM})$. + +??? note "*Proof*:" + + Will be added later. + +## 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 +> +> $$ +> R(\bm{\omega}, \mathbf{u}, \mathbf{v}, \mathbf{w}) = \mathbf{k}(\bm{\omega}, \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u}), +> $$ +> +> for all $\bm{\omega} \in \Gamma(\mathrm{T}^*\mathrm{M})$ and $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM})$. + diff --git a/docs/en/mathematics/differential-geometry/derivatives.md b/docs/en/mathematics/differential-geometry/derivatives.md index e69de29..eefe43b 100644 --- a/docs/en/mathematics/differential-geometry/derivatives.md +++ b/docs/en/mathematics/differential-geometry/derivatives.md @@ -0,0 +1,60 @@ +# Derivatives + +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. + +## Lie derivative + +> *Definition 1*: the **Lie derivative** on a section of a tangent bundle $\mathscr{L}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to \Gamma(\mathrm{TM})$ is a map defined by +> +> $$ +> \mathscr{L}_\mathbf{w} \mathbf{v} = \mathbf{w} \circ \mathbf{v} - \mathbf{v} \circ \mathbf{w} = [\mathbf{w}, \mathbf{v}], +> $$ +> +> for all $\mathbf{w}, \mathbf{v} \in \Gamma(\mathrm{TM})$. + +In which the bracket formulation is also referred to as the Lie bracket. + +> *Proposition 1*: the Lie derivative can be decomposed into +> +> $$ +> \mathscr{L}_\mathbf{w} \mathbf{v} = \mathscr{L}_\mathbf{w}^i \mathbf{v} \partial_i = (w^j \partial_j v^i - v^j \partial_j w^i) \partial_i, +> $$ +> +> for all $\mathbf{w}, \mathbf{v} \in \Gamma(\mathrm{TM})$. + +??? note "*Proof*:" + + Will be added later. + +## Exterior derivative + +> *Definition 2*: the **exterior derivative** $d: \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big) \to \Gamma \big(\bigwedge_{k+1}(\mathrm{T}\mathrm{M}) \big)$ of a $k$-form field, $k \in \mathbb{N}[k \leq n]$ is the $(k+1)$-form field +> +> $$ +> \begin{align*} +> d \bm{\omega} &= d \omega_{|i_1 \dots i_k|} \wedge dx^{i_1} \wedge \dots \wedge dx^{i_k}, \\ +> &= \partial_j \omega_{|i_1 \dots i_k|} dx^j \wedge dx^{i_1} \wedge \dots \wedge dx^{i_k}, +> \end{align*} +> $$ +> +> for all $\bm{\omega} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big)$. + +From the definition of the exterior definition the following results arises. + +> *Theorem 1*: we have that +> +> 1. $\forall\bm{\omega} \in \Gamma \big(\bigwedge_n(\mathrm{T}\mathrm{M}) \big): d \bm{\omega} = \mathbf{0}$, +> 2. $\forall\bm{\omega} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big), k \in \mathbb{N}[k \leq n]: d^2 \bm{\omega} = \mathbf{0}$. + +??? note "*Proof*:" + + Will be added later. + +## Hodge star operator + +> *Definition 3*: the **hodge star operator** $*: \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big) \to \Gamma \big(\bigwedge_{n-k}(\mathrm{T}\mathrm{M}) \big)$ with $k \in \mathbb{N}[k \leq n]$ has the following properties +> +> 1. $\forall \bm{\omega} \in \Gamma \big(\bigwedge_0(\mathrm{T}\mathrm{M}) \big): * \bm{\omega} = \bm{\epsilon}$, +> 2. $* (dx^{i_1} \wedge \dots \wedge dx^{i_k}) = \bm{\epsilon} \lrcorner \mathbf{g}^{-1}(dx^{i_1}) \lrcorner \dots \lrcorner \mathbf{g}^{-1}(dx^{i_k})$, +> +> for all $dx^{i_1} \wedge \dots \wedge dx^{i_k} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big)$ with $\bm{\epsilon}$ the Levi-Civita tensor $\bm{\epsilon} \in \big(\bigwedge_n(\mathrm{T}\mathrm{M}) \big)$ and $\mathbf{g}^{-1}: \Gamma(\mathrm{T}^*\mathrm{M}) \to \Gamma(\mathrm{T}\mathrm{M})$ the [dual metric](). \ No newline at end of file diff --git a/docs/en/mathematics/differential-geometry/linear-connections.md b/docs/en/mathematics/differential-geometry/linear-connections.md index 0af3b84..87cac9e 100644 --- a/docs/en/mathematics/differential-geometry/linear-connections.md +++ b/docs/en/mathematics/differential-geometry/linear-connections.md @@ -1,6 +1,6 @@ # Linear connections -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. +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. > *Definition 1*: a **linear connection** on the fiber bundle $\mathscr{B}$ is a map > @@ -8,11 +8,70 @@ Let $\mathrm{M}$ be a differential manifold with $\dim \mathrm{M} = n \in \mathb > \nabla: \Gamma(\mathrm{TM}) \times \Gamma(\mathscr{B}) \to \Gamma(\mathscr{B}): (\mathbf{v}, \mathbf{T}) \mapsto \nabla_\mathbf{v} \mathbf{T}, > $$ > -> satisfying the following properties, if $f,g \in C^\infty(\mathrm{M})$, $\mathbf{v} \in \mathrm{TM}$ and $\mathbf{T}, \mathbf{S} \in \mathscr{B}$ then +> satisfying the following properties, if $f,g \in C^\infty(\mathrm{M})$, $\mathbf{v} \in \Gamma(\mathrm{TM})$ and $\mathbf{T}, \mathbf{S} \in \Gamma(\mathscr{B})$ then > > 1. $\nabla_{f\mathbf{v}} \mathbf{T} = f \nabla_\mathbf{v} \mathbf{T}$ > 2. $\nabla_\mathbf{v} (f \mathbf{T} + g \mathbf{S}) = (\nabla_\mathbf{v} f) \mathbf{T} + f \nabla_\mathbf{v} \mathbf{T} + (\nabla_\mathbf{v} g) \mathbf{S} + g \nabla_{\mathbf{v}} \mathbf{S}$, > 3. $\nabla_\mathbf{v} f = \mathbf{v} f = \mathbf{k}(df, \mathbf{v})$. -From property 3 it becomes clear that $\nabla_\mathbf{v}$ is an analogue of a directional derivative. +From property 3 it becomes clear that $\nabla_\mathbf{v}$ is an analogue of a directional derivative. The linear connection can also be defined in terms of the cotangent bundle and the dual fiber bundle. +## Covariant derivative + +> *Definition 2*: let $\mathbf{v} = v^i \mathbf{e}_i\in \Gamma(\mathscr{B})$ then the **covariant derivative** on $\mathbf{v}$ is defined as +> +> $$ +> D_k \mathbf{v} \overset{\text{def}}= \nabla_{\partial_k} \mathbf{v} = (\partial_k v^i) \mathbf{e}_i + v^i \Gamma^j_{ik} \mathbf{e}_j = (\partial_k v^i + \Gamma^i_{jk} v^j)\mathbf{e}_i, +> $$ +> +> with formally $\mathbf{k}(\mathbf{\hat e}^j, \nabla_{\partial_k} \mathbf{e}_i) = \Gamma^j_{ik}$ the **linear connection symbols**, in this case $\nabla_{\partial_k} \mathbf{e}_i = \Gamma^j_{ik} \mathbf{e}_j$. + +The covariant derivative can thus be seen as a linear connection for which only the basis is used of the tangent vector. The covariant derivative can also be applied on higher, mixed rank tensors $\mathbf{T} = T^{ij}_k \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k \in \Gamma(\mathscr{B})$ which obtains + +$$ + D_l \mathbf{T} = (\partial_l T^{ij}_k) \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k + T^{ij}_k (\Gamma_{il}^m\mathbf{e}_m) \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k + T^{ij}_k \mathbf{e}_i \otimes (\Gamma^m_{jl} \mathbf{e}_m) \otimes \mathbf{\hat e}^k + T^{ij}_k \mathbf{e}_i \otimes \mathbf{e}_j \otimes (\hat \Gamma^k_{ml} \mathbf{\hat e}^m), +$$ + +with the dual linear connection symbols given by $\mathbf{k}(\nabla_{\partial_k} \mathbf{\hat e}^i, \mathbf{e}_j) = \hat \Gamma^j_{ik}$ with $\nabla_{\partial_k} \mathbf{\hat e}^i = \hat \Gamma^j_{ik} \mathbf{\hat e}^j$. We then have the following proposition such that we can simplify the above expression. + +> *Proposition 1*: let $\Gamma^j_{ik}$ be the linear connection symbols of a covariant derivative and let $\hat \Gamma^j_{ik}$ be the dual linear connection symbols given by $\mathbf{k}(\nabla_{\partial_k} \mathbf{\hat e}^i, \mathbf{e}_j) = \hat \Gamma^j_{ik}$, then we have that +> +> $$ +> \hat \Gamma^j_{ik} = - \Gamma^j_{ik}, +> $$ +> +> for all $(i,j,k) \in \mathbb{N}^3$. + +??? note "*Proof*:" + + Will be added later. + +With the result of proposition 1 we may write + +$$ + D_l \mathbf{T} = (\partial_l T^{ij}_k + \Gamma_{ml}^i T^{mj}_k + \Gamma_{ml}^j T^{im}_k - \Gamma_{kl}^m T^{ij}_m) \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k. +$$ + +### Transformation of linear connection symbols + +Will be added later. + +## Parallel transport + +> *Definition 3*: let $\mathbf{v} \in \Gamma(\mathrm{TM})$, then **parallel transport** of $\mathbf{v}$ occurs along the manifold $\mathrm{M}$ when +> +> $$ +> D_k \mathbf{v} = \mathbf{0}. +> $$ + +For example, a parameterised vector field $\mathbf{v}: x(t) \mapsto \mathbf{v}(x(t)) \in \Gamma(\mathrm{TM})$ is transported parallel if + +$$ + D_t \mathbf{v} = (\partial_k v^i) \dot x^k \partial_i + \Gamma^i_{jk} v^j \partial_i = \mathbf{0}, +$$ + +so $(\partial_k v^i) \dot x^k + \Gamma^i_{jk} v^j = 0$ + +## Contravariant derivative + +Will be added later. \ No newline at end of file diff --git a/docs/en/mathematics/differential-geometry/torsion.md b/docs/en/mathematics/differential-geometry/torsion.md index e69de29..d49bb97 100644 --- a/docs/en/mathematics/differential-geometry/torsion.md +++ b/docs/en/mathematics/differential-geometry/torsion.md @@ -0,0 +1,39 @@ +# Torsion + +Let $\mathrm{M}$ be a differential manifold with $\dim \mathrm{M} = n \in \mathbb{N}$ used throughout the section. Let $\mathrm{TM}$ and $\mathrm{T^*M}$ denote the tangent and cotangent bundle, $V$ and $V^*$ the fiber and dual fiber bundle and $\mathscr{B}$ the tensor fiber bundle. + +## Torsion operator + +> *Definition 1*: the **torsion operator** $\Theta: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to \Gamma(\mathrm{TM})$ is defined as +> +> $$ +> \Theta(\mathbf{u}, \mathbf{v}) = \nabla_\mathbf{u} \mathbf{v} - \nabla_\mathbf{v} \mathbf{u} - \mathscr{L}_\mathbf{u} \mathbf{v}, +> $$ +> +> for all $\mathbf{u}, \mathbf{v} \in \Gamma(\mathrm{TM})$ and $\mathscr{L}$ the [Lie derivative](). + +Using this definition we obtain the following results. + +> *Proposition 1*: the decomposition of the torsion operator results into +> +> $$ +> \mathbf{k}(\bm{\omega}, \Theta(\mathbf{u}, \mathbf{v})) = \omega_i u^j v^k (\Gamma^i_{kj} - \Gamma^i_{jk}), +> $$ +> +> for all $\bm{\omega} \in \Gamma(\mathrm{T}^*\mathrm{M})$ and $\mathbf{u}, \mathbf{v} \in \Gamma(\mathrm{TM})$. + +??? note "*Proof*:" + + Will be added later. + +## Torsion tensor + +As a result of proposition 1 we may view torsion as a locally defined mixed tensor of type $\mathbf{T} \in \mathrm{T}_x \mathrm{M} \otimes \mathrm{T}_x^* \mathrm{M} \otimes \mathrm{T}_x^* \mathrm{M}$. + +> *Definition 2*: the **torsion tensor** $\mathbf{T}: \mathrm{T}_x^* \mathrm{M} \times \mathrm{T}_x \mathrm{M} \times \mathrm{T}_x \mathrm{M} \to \mathbb{K}$ with $x \in \mathrm{M}$ is defined as +> +> $$ +> \mathbf{T}(\bm{\omega}, \mathbf{u}, \mathbf{v}) = \mathbf{k} \big(\bm{\omega}, \Theta(\mathbf{u}, \mathbf{v}) \big), +> $$ +> +> for all $\bm{\omega} \in \mathrm{T}^*_x\mathrm{M}$ and $\mathbf{u}, \mathbf{v} \in \mathrm{T}_x \mathrm{M}$. \ No newline at end of file