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docs/mathematics/differential-geometry/curvature.md
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docs/mathematics/differential-geometry/curvature.md
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# Curvature
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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.
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## Curvature operator
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> *Definition 1*: the **curvature operator** $\Omega: \Gamma(\mathrm{TM})^3 \to \Gamma(\mathrm{TM})$ is defined as
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>
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> $$
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> \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u} = [\nabla_\mathbf{v}, \nabla_\mathbf{w}] \mathbf{u} - \nabla_{[\mathbf{v}, \mathbf{w}]}\mathbf{u},
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> $$
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>
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> for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM})$ with $[\cdot, \cdot]$ denoting the [Lie bracket]().
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It then follows from the definition that the curvature operator $\Omega$ can be decomposed.
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> *Proposition 1*: the decomposition of the curvature operator $\Omega$ relative to a basis $\{\partial_i\}_{i=1}^n$ of $\Gamma(\mathrm{TM})$ results into
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>
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> $$
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> \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u} = v^i w^j [D_i, D_j] u^l \partial_l,
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> $$
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>
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> for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM})$.
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??? note "*Proof*:"
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Will be added later.
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## Curvature tensor
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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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> $$
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> \mathbf{R}(\bm{\omega}, \mathbf{u}, \mathbf{v}, \mathbf{w}) = \mathbf{k}(\bm{\omega}, \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u}),
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> $$
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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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The Riemann curvature defines the curvature of the differential manifold at a certain point $x \in \mathrm{M}$.
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> *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
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>
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> $$
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> \mathbf{R} = R^i_{jkl} \partial_i \otimes dx^j \otimes dx^k \otimes dx^l,
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> $$
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>
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> then we have that its holor is given by
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>
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> $$
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> R^i_{jkl} = \partial_k \Gamma^i_{jl} + \Gamma^m_{jl} \Gamma^i_{mk} - \partial_k \Gamma^i_{jk} - \Gamma^m_{jk} \Gamma^i_{ml},
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> $$
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>
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> for all $(i,j,k,l) \in \{1, \dots, n\}^4$ with $\Gamma^i_{jk}$ denoting the linear connection symbols.
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??? note "*Proof*:"
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Will be added later.
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It may then be observed that $R^i_{jkl} = - R^i_{jlk}$ such that
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$$
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\mathbf{R} = \frac{1}{2} R^i_{jkl} \partial_i \otimes dx^j \otimes (dx^k \wedge dx^l).
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$$
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60
docs/mathematics/differential-geometry/derivatives.md
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docs/mathematics/differential-geometry/derivatives.md
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# Derivatives
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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.
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## Lie derivative
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> *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
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>
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> $$
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> \mathscr{L}_\mathbf{w} \mathbf{v} = \mathbf{w} \circ \mathbf{v} - \mathbf{v} \circ \mathbf{w} = [\mathbf{w}, \mathbf{v}],
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> $$
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>
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> for all $\mathbf{w}, \mathbf{v} \in \Gamma(\mathrm{TM})$.
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In which the bracket formulation is also referred to as the Lie bracket.
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> *Proposition 1*: the Lie derivative can be decomposed into
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>
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> $$
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> \mathscr{L}_\mathbf{w} \mathbf{v} = \mathscr{L}_\mathbf{w}^i \mathbf{v} \partial_i = (w^j \partial_j v^i - v^j \partial_j w^i) \partial_i,
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> $$
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>
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> for all $\mathbf{w}, \mathbf{v} \in \Gamma(\mathrm{TM})$.
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??? note "*Proof*:"
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Will be added later.
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## Exterior derivative
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> *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
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>
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> $$
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> \begin{align*}
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> d \bm{\omega} &= d \omega_{|i_1 \dots i_k|} \wedge dx^{i_1} \wedge \dots \wedge dx^{i_k}, \\
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> &= \partial_j \omega_{|i_1 \dots i_k|} dx^j \wedge dx^{i_1} \wedge \dots \wedge dx^{i_k},
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> \end{align*}
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> $$
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>
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> for all $\bm{\omega} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big)$.
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From the definition of the exterior definition the following results arises.
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> *Theorem 1*: we have that
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>
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> 1. $\forall\bm{\omega} \in \Gamma \big(\bigwedge_n(\mathrm{T}\mathrm{M}) \big): d \bm{\omega} = \mathbf{0}$,
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> 2. $\forall\bm{\omega} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big), k \in \mathbb{N}[k \leq n]: d^2 \bm{\omega} = \mathbf{0}$.
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??? note "*Proof*:"
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Will be added later.
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## Hodge star operator
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> *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
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>
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> 1. $\forall \bm{\omega} \in \Gamma \big(\bigwedge_0(\mathrm{T}\mathrm{M}) \big): * \bm{\omega} = \bm{\epsilon}$,
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> 2. $* (dx^{i_1} \wedge \dots \wedge dx^{i_k}) = \bm{\epsilon} \lrcorner \mathbf{g}^{-1}(dx^{i_1}) \lrcorner \dots \lrcorner \mathbf{g}^{-1}(dx^{i_k})$,
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>
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> 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]().
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# Differential manifolds
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In the following sections of differential geometry we make use of the Einstein summation convention introduced in [vector analysis](/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates/) and $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}.$
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## Definition
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Differential geometry is concerned with *differential manifolds*, smooth continua that are locally Euclidean.
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> *Definition 1*: let $n \in \mathbb{N}$, a $n$-dimensional **differential manifold** is a Hausdorff (T2) space $M$ furnished with a family of smooth diffeomorphisms $\phi_\alpha: \mathscr{D}(\phi_\alpha) \to \mathscr{R}(\phi_\alpha)$ with $\mathscr{D}(\phi_\alpha) \subset\mathrm{M}$ and $\mathscr{R}(\phi_\alpha) \subset E$, with the following axioms
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>
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> 1. $\mathscr{D}(\phi_\alpha)$ is open and $\bigcup_{\alpha \in \mathbb{N}} \mathscr{D}(\phi_\alpha) =\mathrm{M}$,
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> 2. if $\Omega = \mathscr{D}(\phi_\alpha) \cap \mathscr{D}(\phi_\beta) \neq \empty$ then $\phi_\alpha(\Omega), \phi_\beta(\Omega) \subset E$ are open sets and $\phi_\alpha \circ \phi_\beta^{-1}, \phi_\beta \circ \phi_\alpha$ are diffeomorphisms,
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> 3. the atlas $\mathscr{A} = \{(\mathscr{D}(\phi_\alpha), \phi_\alpha)\}$ is maximal.
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>
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> with $E$ a $n$-dimensional [Euclidean space]().
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The last axiom ensures that any chart is tacitly assumed to be already contained in the atlas.
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## Coordinate transformations
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> *Definition 2*: let $p,q \in \mathrm{M}$ be points on the differential manifold and let $\psi: \mathscr{D}(\psi) \to\mathrm{M}: p \mapsto \psi(p) \overset{\text{def}}{=} q$ be a **transformation** from $p$ to $q$ on the manifold, we define two diffeomorphisms
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>
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> $$
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> \phi_\alpha: \mathscr{D}(\phi_\alpha) \to \mathscr{R}(\phi_\alpha): p \mapsto \phi_\alpha(p) \overset{\text{def}}{=} x,
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> $$
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>
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> $$
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> \phi_\beta: \mathscr{D}(\phi_\beta) \to \mathscr{R}(\phi_\beta): q \mapsto \phi_\beta(q) \overset{\text{def}}{=} y,
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> $$
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>
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> with $\mathscr{D}(\phi_{\alpha,\beta}) \subset\mathrm{M}$ and $\mathscr{R}(\phi_{\alpha,\beta}) \subset E$. Then we have a **coordinate transformation** given by
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>
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> $$
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> \phi_{\alpha \beta}^\psi = \phi_\beta \circ \psi \circ \phi_\alpha^{-1}: x \mapsto y,
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> $$
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>
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> then $\phi_{\alpha \beta}^\psi$ is an **active transformation** if $p \neq q$ and $\phi_{\alpha \beta}^\psi$ is a **passive transformation** if $p = q$.
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To clarify the definitions, a passive transformation corresponds only to a descriptive transformation. Whereas an active transformation corresponds to a transformation on the manifold $M$.
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A passive transformation may also be given directly by $\phi_\beta \circ \phi_\alpha: x \mapsto y$ since $\psi = \mathrm{id}$ in this case. Note that the definitions could also have been given by the inverse as the transformations are all diffeomorphisms.
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# Lengths and volumes
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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.
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## Riemannian geometry
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> *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
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>
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> $$
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> \|\mathbf{v}\| = \sqrt{\bm{g}(\mathbf{v},\mathbf{v})}.
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> $$
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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
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$$
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\begin{align*}
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l_{12} &= \int_{t_1}^{t_2} \|\mathbf{\dot v}(t)\| dt, \\
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&= \int_{t_1}^{t_2} \sqrt{\bm{g}(\mathbf{\dot v},\mathbf{\dot v})} dt, \\
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&= \int_{t_1}^{t_2} \sqrt{g_{ij} \dot v^i \dot v^j} dt,
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\end{align*}
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$$
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with $\mathbf{\dot v} = \dot v^i \partial_i \in \Gamma(\mathrm{TM})$.
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> *Definition 2*: the volume $V$ span by the vectors $\{\mathbf{v}_i\}_{i=1}^n$ in $\Gamma(\mathrm{TM})$ is defined by
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>
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> $$
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> V = \bm{\epsilon}(\mathbf{v}_1, \dots, \mathbf{v}_n) = \sqrt{g} \bm{\mu}(\mathbf{v}_1, \dots, \mathbf{v}_n),
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> $$
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>
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> with $\bm{\epsilon}$ the unique unit volume form.
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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
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$$
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V = \int_S \bm{\epsilon} = \int_S \sqrt{g} dx^1 \dots dx^k.
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$$
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It follows that for $k=1$
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$$
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\int_S \bm{\epsilon} = \int_S \sqrt{\bm{g}}.
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$$
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## Finsler geometry
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Will be added later.
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146
docs/mathematics/differential-geometry/linear-connections.md
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# Linear connections
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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.
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> *Definition 1*: a **linear connection** on the fiber bundle $\mathscr{B}$ is a map
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>
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> $$
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> \nabla: \Gamma(\mathrm{TM}) \times \Gamma(\mathscr{B}) \to \Gamma(\mathscr{B}): (\mathbf{v}, \mathbf{T}) \mapsto \nabla_\mathbf{v} \mathbf{T},
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> $$
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>
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> 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
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>
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> 1. $\nabla_{f\mathbf{v}} \mathbf{T} = f \nabla_\mathbf{v} \mathbf{T}$
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> 2. $\nabla_\mathbf{v} (f \mathbf{T} + g \mathbf{S}) = (\nabla_\mathbf{v} f) \mathbf{T} + f \nabla_\mathbf{v} \mathbf{T} + (\nabla_\mathbf{v} g) \mathbf{S} + g \nabla_{\mathbf{v}} \mathbf{S}$,
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> 3. $\nabla_\mathbf{v} f = \mathbf{v} f = \mathbf{k}(df, \mathbf{v})$.
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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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>
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> $$
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> D_k \mathbf{v} \overset{\text{def}}= \nabla_{\partial_k} \mathbf{v} = (\partial_k v^i) \mathbf{e}_i + v^i \Gamma^j_{ik} \mathbf{e}_j = (\partial_k v^i + \Gamma^i_{jk} v^j)\mathbf{e}_i,
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> $$
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>
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> with formally $\mathbf{k}(\mathbf{\hat e}^j, \nabla_{\partial_k} \mathbf{e}_i) = \Gamma^j_{ik}$ the **linear connection symbols**, in this case $\nabla_{\partial_k} \mathbf{e}_i = \Gamma^j_{ik} \mathbf{e}_j$.
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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
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$$
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D_l \mathbf{T} = (\partial_l T^{ij}_k) \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k + T^{ij}_k (\Gamma_{il}^m\mathbf{e}_m) \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k + T^{ij}_k \mathbf{e}_i \otimes (\Gamma^m_{jl} \mathbf{e}_m) \otimes \mathbf{\hat e}^k + T^{ij}_k \mathbf{e}_i \otimes \mathbf{e}_j \otimes (\hat \Gamma^k_{ml} \mathbf{\hat e}^m),
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$$
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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.
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> *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
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>
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> $$
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> \hat \Gamma^j_{ik} = - \Gamma^j_{ik},
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> $$
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>
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> for all $(i,j,k) \in \mathbb{N}^3$.
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??? note "*Proof*:"
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Will be added later.
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With the result of proposition 1 we may write
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$$
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D_l \mathbf{T} = (\partial_l T^{ij}_k + \Gamma_{ml}^i T^{mj}_k + \Gamma_{ml}^j T^{im}_k - \Gamma_{kl}^m T^{ij}_m) \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k.
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$$
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### Transformation of linear connection symbols
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Will be added later.
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## Intrinsic derivative
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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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> $$
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> D_t \mathbf{v}(t) = \nabla_{\dot\gamma} \mathbf{v}(t),
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> $$
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>
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> for all $t \in \mathscr{D}(\gamma)$.
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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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$$
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\begin{align*}
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\nabla_{\dot\gamma} \mathbf{v}(t) &= \dot \gamma^i \nabla_{\partial_i} (v^j \partial_j), \\
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&= \dot \gamma^i \big((\partial_i v^j) \partial_j + v^j \Gamma_{ji}^k \partial_k \big), \\
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&= (\dot \gamma^i \partial_i v^j + \dot \gamma^i \Gamma^j_{ki}v^k) \partial_j, \\
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&= (\dot v^j + \Gamma^j_{ki} v^k \dot \gamma^i) \partial_j,
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\end{align*}
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$$
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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
|
||||
|
||||
$$
|
||||
\dot v^j + \Gamma^j_{ki} v^k \dot \gamma^i = 0,
|
||||
$$
|
||||
|
||||
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
|
||||
>
|
||||
> $$
|
||||
> \mathscr{L} = \|\dot \gamma\|^2,
|
||||
> $$
|
||||
>
|
||||
> 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
|
||||
|
||||
Will be added later.
|
||||
136
docs/mathematics/differential-geometry/tangent-spaces.md
Normal file
136
docs/mathematics/differential-geometry/tangent-spaces.md
Normal file
|
|
@ -0,0 +1,136 @@
|
|||
# Tangent spaces
|
||||
|
||||
Let $\mathrm{M}$ be a differential manifold with $\dim \mathrm{M} = n \in \mathbb{N}$ used throughout the section.
|
||||
|
||||
## Definition
|
||||
|
||||
> *Definition 1*: let $f \in C^{\infty}(\mathrm{M})$ with $C^{\infty}$ the class of [smooth functions]() and $M$ a differential manifold. A derivation of $f$ at $x \in \mathrm{M}$ is defined as a linear map $\mathbf{v}_x: C^\infty(\mathrm{M}) \to \mathbb{K}$ that satisfies
|
||||
>
|
||||
> $$
|
||||
> \forall f,g \in C^{\infty}(\mathrm{M}): \mathbf{v}_x(f g) = (\mathbf{v}_xf) g + f (\mathbf{v}_x g).
|
||||
> $$
|
||||
>
|
||||
> Let $\mathrm{T}_x\mathrm{M}$ be the set of all derivations at $x$ such that $\mathbf{v}_x \in \mathrm{T}_x\mathrm{M}$. With $\mathrm{T}_x\mathrm{M}$ denoted as the **tangent space** at $x$.
|
||||
|
||||
We may think of the tangent space at a point $x \in \mathrm{M}$ as a space attached to $x$ on the differential manifold $M$.
|
||||
|
||||
## Properties of tangent spaces
|
||||
|
||||
> *Theorem 1*: let $M$ be a differential manifold and let $x \in \mathrm{M}$, the tangent space $\mathrm{T}_x\mathrm{M}$ is a vector space.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
Thus, the tangent space is a vector space attached to $x \in \mathrm{M}$ on the differential manifold. It follows that its vectors have interesting properties.
|
||||
|
||||
> *Theorem 2*: let $M$ be a differential manifold, let $x \in \mathrm{M}$ and let $\mathbf{v}_x \in \mathrm{T}_x\mathrm{M}$, then we have that
|
||||
>
|
||||
> $$
|
||||
> \forall f \in C^{\infty}(\mathrm{M}): \mathbf{v}_x f = v^i \partial_i f(x),
|
||||
> $$
|
||||
>
|
||||
> such that $\mathbf{v}_x = v^i \partial_i \in \mathrm{T}_x\mathrm{M}$ is denoted as a **tangent vector** in the tangent space $\mathrm{T}_x\mathrm{M}$.
|
||||
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
Theorem 2 adds the notion of tangent vectors to the explanation of the tangent space. The tangent space at a point on the manifold thus represents the space of tangent vectors.
|
||||
|
||||
> *Proposition 1*: let $M$ be a differential manifold of $\dim\mathrm{M} = n \in \mathbb{N}$. The tangent space $\mathrm{T}_x\mathrm{M}$ has dimension $n$ such that
|
||||
>
|
||||
> $$
|
||||
> \forall x \in \mathrm{M}: \dim \mathrm{T}_x\mathrm{M} = \dim\mathrm{M}
|
||||
> $$
|
||||
>
|
||||
> and is span by the vector basis $\{\partial_i\}_{i=1}^n$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
Proposition 1 states that the tangent space is of the same dimension as the manifold and its basis are partial derivative operators. In the context of the [covariant basis](), this definition of the basis leaves out the coordinate map, but is in fact equivalent to the covariant basis.
|
||||
|
||||
As a last step in the explanation, we may think of the 2 dimensional surface of a sphere, which may define a differential manifold $M$. The tangent space at a point $x \in \mathrm{M}$ on the surface of the sphere may then be compared to the tangent plane to the sphere attached at point $x \in \mathrm{M}$. The catch is that the 3 dimensional space necessary to understand this construction exists only in our imagination and not in the mathematical construct.
|
||||
|
||||
## Tangent bundle
|
||||
|
||||
> *Definition 2*: let $M$ be a differential manifold, the collection of tangent spaces $\mathrm{T}_x\mathrm{M}$ for all $x \in \mathrm{M}$ define the **tangent bundle** as
|
||||
>
|
||||
> $$
|
||||
> \mathrm{TM} = \bigcup_{x \in \mathrm{M}} \mathrm{T}_x\mathrm{M}.
|
||||
> $$
|
||||
|
||||
In particular, we may think of the tangent bundle $\mathrm{TM}$ as a subspace $\mathrm{TM} \subset V$ of the fiber bundle $V$ for a differential manifold. With the special properties given in theorem 2 and proposition 1.
|
||||
|
||||
The connection of each tangent vector to its base point may be formalised with the projection map $\pi$ which in this case is given by
|
||||
|
||||
$$
|
||||
\pi: \mathrm{TM} \to\mathrm{M}: (x, \mathbf{v}) \mapsto \pi(x, \mathbf{v}) \overset{\text{def}}{=} x,
|
||||
$$
|
||||
|
||||
and its inverse
|
||||
|
||||
$$
|
||||
\pi^{-1}:\mathrm{M} \to \mathrm{TM}: x \mapsto \pi^{-1}(x) \overset{\text{def}}{=} \mathrm{T}_x\mathrm{M}.
|
||||
$$
|
||||
|
||||
> *Definition 3*: a vector field $\mathbf{v}$ on a differential manifold $M$ is a section
|
||||
>
|
||||
> $$
|
||||
> \mathbf{v} \in \Gamma(\mathrm{TM}),
|
||||
> $$
|
||||
>
|
||||
> of the tangent bundle $\mathrm{TM}$.
|
||||
|
||||
## Cotangent spaces
|
||||
|
||||
> *Definition 4*: let $M$ be a differential manifold and $\mathrm{T}_x\mathrm{M}$ the tangent space at $x \in \mathrm{M}$. We define the **cotangent space** $\mathrm{T}_x^*\mathrm{M}$ as the dual space of $\mathrm{T}_x\mathrm{M}$
|
||||
>
|
||||
> $$
|
||||
> \mathrm{T}_x^*\mathrm{M} = (\mathrm{T}_x\mathrm{M})^*.
|
||||
> $$
|
||||
>
|
||||
> Then every element $\bm{\omega}_x \in \mathrm{T}_x^*\mathrm{M}$ is a linear map $\bm{\omega}_x: \mathrm{T}_x\mathrm{M} \to \mathbb{K}$ denoted as the **cotangent vector**.
|
||||
|
||||
This definition is a logical consequence of the notion of the [dual vector space](). It then also follows that the dual cotangent space is isomorphic to the tangent space at a point $x \in \mathrm{M}$.
|
||||
|
||||
> *Theorem 3*: let $\mathrm{M}$ be a differential manifold of $\dim \mathrm{M} = n \in \mathbb{N}$, then we have that for every $x \in \mathrm{M}$ the basis $\{dx^i\}_{i=1}^n$ of $\mathrm{T}_x^*\mathrm{M}$ is uniquely determined by
|
||||
>
|
||||
> $$
|
||||
> dx^i(\partial_j) = \delta^i_j,
|
||||
> $$
|
||||
>
|
||||
> for each basis $\{\partial_j\}_{j=1}^n$ in $\mathrm{T}_x\mathrm{M}$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
The proof follows directly from theorem 1 in [dual vector spaces]().
|
||||
|
||||
The choice of $dx^i$ can be explained by taking the differential $df = \partial_i f dx^i \in \mathrm{T}_x^*\mathrm{M}$ with $f \in C^\infty(\mathrm{M})$. Then if we take
|
||||
|
||||
$$
|
||||
\mathbf{k}_x(df, \mathbf{v}) = \mathbf{k}(\partial_i f dx^i, v^j \partial_j) = v^j \partial_i f \mathbf{k}(dx^i, \partial_j) = v^j \partial_i f \delta^i_j = v^i \partial_i f = \mathbf{v} f,
|
||||
$$
|
||||
|
||||
with $\mathbf{k}_x: \mathrm{T}_x^*\mathrm{M} \times \mathrm{T}_x\mathrm{M} \to \mathbb{K}$ the Kronecker tensor at $x \in \mathrm{M}$. Which shows that defining the basis of the cotangent space as differentials corresponds with respect to the basis of the tangent space.
|
||||
|
||||
So, a cotangent vector $\bm{\omega}_x \in \mathrm{T}_x^*\mathrm{M}$ may be decomposed into
|
||||
|
||||
$$
|
||||
\bm{\omega}_x = \omega_i dx^i.
|
||||
$$
|
||||
|
||||
In the context of the [contravariant basis](), this definition of the basis leaves out the coordinate map, but is in fact equivalent to the contravariant basis.
|
||||
|
||||
## Cotangent bundle
|
||||
|
||||
> *Definition 5*: let $M$ be a differential manifold, the collection of cotangent spaces $\mathrm{T}_x^*\mathrm{M}$ for all $x \in \mathrm{M}$ define the **cotangent bundle** as
|
||||
>
|
||||
> $$
|
||||
> \mathrm{T^*M} = \bigcup_{x \in \mathrm{M}} \mathrm{T}_x^*\mathrm{M}.
|
||||
> $$
|
||||
|
||||
Thus, we may think of the cotangent bundle $\mathrm{T^*M}$ as a subspace $\mathrm{T^*M} \subset V^*$ of the dual fiber bundle $V^*$ for a differential manifold.
|
||||
39
docs/mathematics/differential-geometry/torsion.md
Normal file
39
docs/mathematics/differential-geometry/torsion.md
Normal file
|
|
@ -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 $\mathbf{u}, \mathbf{v} \in \mathrm{T}_x \mathrm{M}$.
|
||||
35
docs/mathematics/differential-geometry/transformations.md
Normal file
35
docs/mathematics/differential-geometry/transformations.md
Normal file
|
|
@ -0,0 +1,35 @@
|
|||
# Transformations
|
||||
|
||||
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.
|
||||
|
||||
## Push forward and pull back
|
||||
|
||||
> *Definition 1*: let $\mathrm{M}, \mathrm{N}$ be two differential manifolds with $\dim \mathrm{N} \geq \dim \mathrm{M}$ and let $\psi: \mathrm{M} \to \mathrm{N}$ be the diffeomorphism between the manifolds. Then we define the **pull back** $\psi^*$ and **push forward** $\psi_*$ operators, such that for $\mathbf{v} \in \mathrm{T}_x \mathrm{M}$ and $\bm{\omega} \in \mathrm{T}_{\psi(x)}^* \mathrm{M}$ we have
|
||||
>
|
||||
> $$
|
||||
> \mathbf{k}_x(\psi^* \bm{\omega}, \mathbf{v}) = \mathbf{k}_{\psi(x)}(\bm{\omega}, \psi_* \mathbf{v}),
|
||||
> $$
|
||||
>
|
||||
> for all $x \in \mathrm{M}$.
|
||||
|
||||
Which indicates the proper separation between the elements of both spaces.
|
||||
|
||||
## Basis transformation
|
||||
|
||||
Let $\psi: \mathscr{D}(\mathrm{M}) \to \mathrm{M}: x \mapsto \psi(x) \overset{\text{def}}{=} \overline{x}$ be an active coordinate transformation from a point $x$ to a point $\overline{x}$ on $\mathrm{M}$. Then we have a basis $\{\partial_i\}_{i=1}^n \subset \mathrm{T}_x\mathrm{M}$ for the tangent space $\mathrm{T}_x\mathrm{M}$ at $x$ and a basis $\{\overline{\partial_i}\}_{i=1}^n \subset \mathrm{T}_{\overline{x}}\mathrm{M}$ for the tangent space $\mathrm{T}_{\overline{x}}\mathrm{M}$ at $\overline{x}$. Which are related by
|
||||
|
||||
$$
|
||||
\partial_i = J^j_i \overline{\partial_j} = \partial_i \psi^j(x) \overline{\partial_j},
|
||||
$$
|
||||
|
||||
with $J^j_i = \partial_i \psi^j(x)$ the [Jacobian]() at $x \in \mathrm{M}$. For it to make sense, it helps to change notation to
|
||||
|
||||
$$
|
||||
\frac{\partial}{\partial x_i} = \frac{\partial \overline{x}^j}{\partial x_i} \frac{\partial}{\partial \overline{x}_j} = \frac{\partial \psi^j}{\partial x_i} \frac{\partial}{\partial \overline{x}_j}.
|
||||
$$
|
||||
|
||||
Similarly, we have a basis $\{dx^i\}_{i=1}^n \subset \mathrm{T}_x^*\mathrm{M}$ for the cotangent space $\mathrm{T}_x\mathrm{M}$ at $x$ and a basis $\{d\overline{x}^i\}_{i=1}^n \subset \mathrm{T}_{\overline{x}}^*\mathrm{M}$ for the cotangent space $\mathrm{T}_{\overline{x}}^*\mathrm{M}$ at $\overline{x}$. Which are related by
|
||||
|
||||
$$
|
||||
d\overline{x}^i = J^i_j dx^j = \partial_j \psi^i(x) dx^j.
|
||||
$$
|
||||
Loading…
Add table
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Reference in a new issue