From 2983b6afc44ed12954b821be99a60191669e32f9 Mon Sep 17 00:00:00 2001 From: Luc Date: Thu, 20 Jun 2024 20:09:38 +0200 Subject: [PATCH] Updated and added some parts to differential geometry and tensors. --- .../differential-geometry/curvature.md | 27 ++++++++++- .../lengths-and-volumes.md | 47 +++++++++++++++++++ .../linear-connections.md | 2 +- .../tensors/tensor-formalism.md | 9 ++-- .../linear-algebra/tensors/volume-forms.md | 2 +- 5 files changed, 80 insertions(+), 7 deletions(-) create mode 100644 docs/en/mathematics/differential-geometry/lengths-and-volumes.md diff --git a/docs/en/mathematics/differential-geometry/curvature.md b/docs/en/mathematics/differential-geometry/curvature.md index e285ffb..a100c4e 100644 --- a/docs/en/mathematics/differential-geometry/curvature.md +++ b/docs/en/mathematics/differential-geometry/curvature.md @@ -28,7 +28,7 @@ It then follows from the definition that the curvature operator $\Omega$ can be ## Curvature tensor -> *Definition 2*: the Riemann curvature tensor $\mathbf{R}: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}$ 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 > > $$ > \mathbf{R}(\bm{\omega}, \mathbf{u}, \mathbf{v}, \mathbf{w}) = \mathbf{k}(\bm{\omega}, \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u}), @@ -36,3 +36,28 @@ It then follows from the definition that the curvature operator $\Omega$ can be > > for all $\bm{\omega} \in \Gamma(\mathrm{T}^*\mathrm{M})$ and $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM})$. +The Riemann curvature defines the curvature of the differential manifold at a certain point $x \in \mathrm{M}$. + +> *Proposition 2*: let $\mathbf{R}: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}$ be the Riemann curvature tensor, with its decomposition given by +> +> $$ +> \mathbf{R} = R^i_{jkl} \partial_i \otimes dx^j \otimes dx^k \otimes dx^l, +> $$ +> +> then we have that its holor is given by +> +> $$ +> R^i_{jkl} = \partial_k \Gamma^i_{jl} + \Gamma^m_{jl} \Gamma^i_{mk} - \partial_k \Gamma^i_{jk} - \Gamma^m_{jk} \Gamma^i_{ml}, +> $$ +> +> for all $(i,j,k,l) \in \{1, \dots, n\}^4$ with $\Gamma^i_{jk}$ denoting the linear connection symbols. + +??? note "*Proof*:" + + Will be added later. + +It may then be observed that $R^i_{jkl} = - R^i_{jlk}$ such that + +$$ + \mathbf{R} = \frac{1}{2} R^i_{jkl} \partial_i \otimes dx^j \otimes (dx^k \wedge dx^l). +$$ \ No newline at end of file diff --git a/docs/en/mathematics/differential-geometry/lengths-and-volumes.md b/docs/en/mathematics/differential-geometry/lengths-and-volumes.md new file mode 100644 index 0000000..c4851d9 --- /dev/null +++ b/docs/en/mathematics/differential-geometry/lengths-and-volumes.md @@ -0,0 +1,47 @@ +# Lengths and volumes + +Let $\mathrm{M}$ be a differential manifold with $\dim \mathrm{M} = n \in \mathbb{N}$ used throughout the section. Let $\mathrm{TM}$ and $\mathrm{T^*M}$ denote the tangent and cotangent bundle, $V$ and $V^*$ the fiber and dual fiber bundle and $\mathscr{B}$ the tensor fiber bundle. + +## Riemannian geometry + +> *Definition 1*: the length of a vector $\mathbf{v} \in \Gamma(\mathrm{TM})$ is defined by the norm $\|\cdot\|$ induced by the inner product $\bm{g}$ such that +> +> $$ +> \|\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}}. +$$ + +## Finsler geometry + +Will be added later. \ No newline at end of file diff --git a/docs/en/mathematics/differential-geometry/linear-connections.md b/docs/en/mathematics/differential-geometry/linear-connections.md index 4e990a4..17dd86a 100644 --- a/docs/en/mathematics/differential-geometry/linear-connections.md +++ b/docs/en/mathematics/differential-geometry/linear-connections.md @@ -124,7 +124,7 @@ One may interpret a geodesic as a generalization of the notion of a straight lin > *Proposition 2*: let $\gamma: \mathscr{D}(\gamma) \to M: t \mapsto \gamma(t)$ be a smooth curve on the manifold parameterized by an open interval $\mathscr{D}(\gamma) \subset \mathbb{R}$ and let $\mathscr{L}$ be the Lagrangian defined by > > $$ -> \mathscr{L} = \|\dot \gamma\|^2 = g_{ij} \dot \gamma^i \dot \gamma^j, +> \mathscr{L} = \|\dot \gamma\|^2, > $$ > > for all $t \in \mathscr{D}(\gamma)$. By demanding [Hamilton's principle]() we obtain the geodesic equations diff --git a/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md b/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md index ca22c73..0b17f59 100644 --- a/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md +++ b/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md @@ -135,11 +135,12 @@ We have from theorem 2 that the outer product of two tensors yields another tens ## Inner product -> *Definition 5*: a **pseudo inner product** on $V$ is a nondegenerate bilinear mapping $\bm{g}: V \times V \to \mathbb{K}$ which satisfies +> *Definition 5*: an **inner product** on $V$ is a bilinear mapping $\bm{g}: V \times V \to \mathbb{K}$ which satisfies > -> 1. for all $\mathbf{u} \in V \backslash \{\mathbf{0}\} \exists \mathbf{v} \in V: \; \bm{g}(\mathbf{u},\mathbf{v}) \neq 0$, -> 2. for all $\mathbf{u}, \mathbf{v} \in V: \; \bm{g}(\mathbf{u}, \mathbf{v}) = \overline{\bm{g}}(\mathbf{v}, \mathbf{u})$, -> 3. for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $\lambda, \mu \in \mathbb{K}: \;\bm{g}(\mathbf{u}, \lambda \mathbf{v} + \mu \mathbf{w}) = \lambda \bm{g}(\mathbf{u}, \mathbf{v}) + \mu \bm{g}(\mathbf{u}, \mathbf{w}).$ +> 1. for all $\mathbf{u}, \mathbf{v} \in V: \; \bm{g}(\mathbf{u}, \mathbf{v}) = \overline{\bm{g}}(\mathbf{v}, \mathbf{u}),$ +> 2. for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $\lambda, \mu \in \mathbb{K}: \;\bm{g}(\mathbf{u}, \lambda \mathbf{v} + \mu \mathbf{w}) = \lambda \bm{g}(\mathbf{u}, \mathbf{v}) + \mu \bm{g}(\mathbf{u}, \mathbf{w}),$ +> 3. for all $\mathbf{u} \in V\backslash \{\mathbf{0}\}: \bm{g}(\mathbf{u},\mathbf{u}) > 0,$ +> 4. for $\mathbf{u} = \mathbf{0} \iff \bm{g}(\mathbf{u},\mathbf{u}) = 0.$ It may be observed that $\bm{g} \in \mathscr{T}_2^0$. Unlike the Kronecker tensor, the existence of an inner product is never implied. diff --git a/docs/en/mathematics/linear-algebra/tensors/volume-forms.md b/docs/en/mathematics/linear-algebra/tensors/volume-forms.md index ec09040..99530be 100644 --- a/docs/en/mathematics/linear-algebra/tensors/volume-forms.md +++ b/docs/en/mathematics/linear-algebra/tensors/volume-forms.md @@ -102,7 +102,7 @@ for $k \in \mathbb{N}[k < n]$. > \bm{\epsilon} = \sqrt{g} \bm{\mu}, > $$ > -> with $g \overset{\text{def}}{=} |\det (G)|$, the absolute value of the determinant of the [Gram matrix](). +> with $g \overset{\text{def}}{=} \det (G)$, the determinant of the [Gram matrix](). Therefore, if we decompose the Levi-Civita tensor by