Added the first section of differential geometry.
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- 'Second order differential equations': mathematics/ordinary-differential-equations/second-order-ode.md
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- 'Systems of linear differential equations': mathematics/ordinary-differential-equations/systems-of-linear-ode.md
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- 'The Laplace transform': mathematics/ordinary-differential-equations/laplace-transform.md
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- 'Differential geometry':
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- 'Differential manifolds': mathematics/differential-geometry/differential-manifolds.md
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- 'Linear connections': mathematics/differential-geometry/linear-connections.md
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- 'Derivatives': mathematics/differential-geometry/derivatives.md
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- 'Torsion': mathematics/differential-geometry/torsion.md
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- 'Curvature': mathematics/differential-geometry/curvature.md
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- 'Physics':
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- physics/index.md
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docs/en/mathematics/differential-geometry/curvature.md
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docs/en/mathematics/differential-geometry/curvature.md
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docs/en/mathematics/differential-geometry/derivatives.md
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# Differential manifolds
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In the following sections 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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## 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** 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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## Fiber bundles
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(This subsection should probably be moved to a more general setting of manifolds.)
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> *Definition 3*: a **fiber** $V_x$ at a point $x \in \mathrm{M}$ on a manifold is a finite dimensional vector space. With the collection of fibers $V_x$ for all $x \in \mathrm{M}$ define the **fiber bundle** as
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>
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> $$
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> V = \bigcup_{x \in \mathrm{M}} V_x.
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> $$
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Then by definition we have the projection map $\pi$ given by
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$$
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\pi: V \to\mathrm{M}: (x,\mathbf{v}) \mapsto \pi(x, \mathbf{v}) \overset{\text{def}}{=} x,
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$$
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and its inverse
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$$
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\pi^{-1}:\mathrm{M} \to V: x \mapsto \pi(x) \overset{\text{def}}{=} V_x.
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$$
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Similarly, a dual fiber $V_x^*$ may be defined for $x \in \mathrm{M}$, with its fiber bundle defined by
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$$
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V^* = \bigcup_{x \in \mathrm{M}} V_x^*.
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$$
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> *Definition 4*: a **tensor fiber** $\mathscr{B}_x$ at a point $x \in \mathrm{M}$ on a manifold is defined as
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>
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> $$
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> \mathscr{B}_x = \bigcup_{p,q \in \mathbb{N}} \mathscr{T}^p_q(V_x).
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> $$
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>
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> With the collection of tensor fibers $\mathscr{B}_x$ for all $x \in \mathrm{M}$ define the **tensor fiber bundle** as
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>
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> $$
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> \mathscr{B} = \bigcup_{x \in \mathrm{M}} \mathscr{B}_x.
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> $$
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Then for a point $x \in \mathrm{M}$ we have a tensor $\mathbf{T} \in \mathscr{B}_x$ such that
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$$
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\mathbf{T} = T^{ij}_k \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k,
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$$
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with $T^{ij}_k \in \mathbb{K}$ holors of $\mathbf{T}$. Furthermore, $\{\mathbf{e}_i\}_{i=1}^n$ a basis of $V_x$ and $\{\mathbf{\hat e}^i\}_{i=1}^n$ a basis of $V_x^*$.
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> *Definition 5*: a tensor field $\mathbf{T}$ on a manifold $M$ is a [section]()
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>
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> $$
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> \mathbf{T} \in \Gamma(\mathrm{M}, \mathscr{B}),
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> $$
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>
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> of the tensor fiber bundle $\mathscr{B}$.
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Therefore, a tensor field assigns a tensor fiber (or tensor) to each point on a section of the manifold. These tensors may vary smoothly along the section of the manifold.
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## Tangent bundles
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> *Definition 6*: 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
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>
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> $$
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> \forall f,g \in C^{\infty}(\mathrm{M}): \mathbf{v}_x(f g) = (\mathbf{v}_xf) g + f (\mathbf{v}_x g).
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> $$
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>
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> 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 $x$.
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We may think of the tangent space at a point $x \in \mathrm{M}$ as a space attached to $x$ on the differential manifold $M$.
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> *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.
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??? note "*Proof*:"
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Will be added later.
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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.
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> *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
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>
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> $$
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> \forall f \in C^{\infty}(\mathrm{M}): \mathbf{v}_x f = v^i \partial_i f(x),
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> $$
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>
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> 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 $\mathrm{T}_x\mathrm{M}$.
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??? note "*Proof*:"
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Will be added later.
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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.
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> *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
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>
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> $$
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> \forall x \in \mathrm{M}: \dim \mathrm{T}_x\mathrm{M} = \dim\mathrm{M}
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> $$
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>
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> and is span by the vector basis $\{\partial_i\}_{i=1}^n$.
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??? note "*Proof*:"
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Will be added later.
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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.
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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.
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> *Definition 7*: 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
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>
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> $$
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> \mathrm{TM} = \bigcup_{x \in \mathrm{M}} \mathrm{T}_x\mathrm{M}.
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> $$
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In particular, we may think of the tangent bundle $\mathrm{TM}$ as a subspace $\mathrm{TM} \subset V$ of the vector bundle $V$ for a differential manifold. With the special properties given in theorem 2 and proposition 1.
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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
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$$
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\pi: \mathrm{TM} \to\mathrm{M}: (x, \mathbf{v}) \mapsto \pi(x, \mathbf{v}) \overset{\text{def}}{=} x,
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$$
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and its inverse
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$$
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\pi^{-1}:\mathrm{M} \to \mathrm{TM}: x \mapsto \pi^{-1}(x) \overset{\text{def}}{=} \mathrm{T}_x\mathrm{M}.
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$$
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> *Definition 8*: a vector field $\mathbf{v}$ on a differential manifold $M$ is a section
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>
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> $$
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> \mathbf{v} \in \Gamma(\mathrm{TM}),
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> $$
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>
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> of the tangent bundle $\mathrm{TM}$.
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## Cotangent bundles
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> *Definition 9*: 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 $\mathrm{T}_x\mathrm{M}$
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>
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> $$
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> \mathrm{T}_x^*\mathrm{M} = (\mathrm{T}_x\mathrm{M})^*.
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> $$
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>
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> 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**.
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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 $x \in \mathrm{M}$.
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> *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
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>
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> $$
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> dx^i(\partial_j) = \delta^i_j,
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> $$
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>
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> for each basis $\{\partial_j\}_{j=1}^n$ in $\mathrm{T}_x\mathrm{M}$.
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??? note "*Proof*:"
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The proof follows directly from theorem 1 in [dual vector spaces]().
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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
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$$
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\mathbf{k}_x(df, \mathbf{v}) = \mathbf{k}(\partial_i f dx^i, v^j \partial_j) = v^j \partial_i f \mathbf{k}(dx^i, \partial_j) = v^j \partial_i f \delta^i_j = v^i \partial_i f = \mathbf{v} f,
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$$
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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.
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So, a cotangent vector $\bm{\omega}_x \in \mathrm{T}_x^*\mathrm{M}$ may be decomposed into
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$$
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\bm{\omega}_x = \omega_i dx^i.
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$$
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## Push forward and pull back
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> *Definition 10*: 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
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>
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> $$
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> \mathbf{k}_x(\psi^* \bm{\omega}, \mathbf{v}) = \mathbf{k}_{\psi(x)}(\bm{\omega}, \psi_* \mathbf{v}),
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> $$
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>
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> for all $x \in \mathrm{M}$.
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Which indicates the proper separation between the elements of both spaces.
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> with $p, q \in \mathbb{N}$. Tensors are collectively denoted as
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>
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> $$
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> \mathbf{T} = \underbrace{V \otimes \dots \otimes V}_p \otimes \underbrace{V^* \otimes \dots \otimes V^*}_q = \mathscr{T}_q^p(V),
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> \mathbf{T} \in \underbrace{V \otimes \dots \otimes V}_p \otimes \underbrace{V^* \otimes \dots \otimes V^*}_q = \mathscr{T}_q^p(V),
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> $$
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>
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> with $\mathscr{T}_0^0(V) = \mathbb{K}$.
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# Tensor symmetries
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We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n,$ a pseudo inner product $\bm{g}$ on $V$ and a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}.$
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We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n,$ a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}$ and a pseudo inner product $\bm{g}$ on $V.$
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## Symmetric tensors
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We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n,$ a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}_{i=1}^n$ and a pseudo inner product $\bm{g}$ on $V.$
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## n forms
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## n-forms
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> *Definition 1*: let $\bm{\mu} \in \bigwedge_n(V) \backslash \{\mathbf{0}\}$, if
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>
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From proposition 2 it may also be observed that within a geometrical context the oriented volume form may represent the area of a parallelogram in $n=2$ or the volume of a parallelepiped in $n=3$, span by its basis.
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## n - k forms
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## (n - k)-forms
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> *Definition 3*: let $(V, \bm{\mu})$ be a vector space with an oriented volume form and let $\mathbf{u}_1, \dots, \mathbf{u}_k \in V$ with $k \in \mathbb{N}[k < n]$. Let the $(n-k)$-form $\bm{\mu} \lrcorner \mathbf{u}_1 \lrcorner \dots \lrcorner \mathbf{u}_k \in \bigwedge_{n-k}(V)$ be defined as
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>
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