Adapted from Lee’s ISM.
Smooth manifolds
A topological manifold is hausdorff, second-countable and locally euclidean.
Smooth maps
\(f : M\to\mathbb{R}^K\) is a smooth function if for every \(p \in M\), there exist a smooth chart \((U,\varphi)\) for M whose domain contains p and such that the composite function \(f \circ \phi^{-1}\) is smooth on the open subset \(\hat{U} = \phi(U) \subseteq \mathbb{R}^n\).
Cotangent bundle
The cotangent bundle is the dual space to the tangent bundle. i.e. the linear functionals which take in tangent vectors.
Covectors
For a given vector space, a covector is a real-valued linear functional on V. The vector space of covectors on \(V\) form its dual space.
The basis of the dual space are the covectors \(\epsilon^i(E_j) = \delta^i_j\).
While vectors are expanded in terms of their bases like \(v^iE_i\), covectors are expanded as \(\omega_i\varepsilon^i\)
The dual map of a linear map \(A : V\to W\) is defined by
\[ (A^*\omega)(v) = \omega(Av) \] for \(\omega \in W^*, v\in V\).
It can be shown that ‘*’ operator is a contravariant functor from the category of real vector spaces to itself.
Second dual space
For each vector space, there is a natural, basis-independent map \(\xi: V \to V^{**}\), defined as follows. For each vector \(v \in V\), define a linear functional \(\xi(v) : V^{ * } \to \mathbb{R}\) by
\[ \xi(v)(\omega) = \omega(v) \] for \(\omega \in V^*\)
Recall the following elementary result on second dual spaces.
For any finite-dimensional vector space V, the map \(\xi: V\to V^{**}\) is an isomorphism
This proposition allows us to denote the the application of a covector t oa vector by either \(\langle \omega, v \rangle\) or \(\langle \omega, v \rangle\).
Tangent covectors on manifolds
The cotangent space at p, is the dual space to \(T_p M\), denoted \(T^*_p M\). Given smooth local coordinates \((x^i)\), the coordinate basis \((\partial_i|_p)\) gives rise to a dual basis for \(T^*_p M\), which will be denoted as \((dx^i|_p)\). Thus any such covector can be written uniquely as \(\omega = \omega_i dx^i|_p\), where \(\omega_i = \omega(\partial_i|_p)\).
Recall from the discussion of tangent spaces that two different smooth coordinates transform into one another by the following transformation of their coordinate vector fields,
\[ \pdv{}{x^i}\bigg\rvert_p = \pdv{\tilde{x}^j}{x^i}(p) \pdv{}{\tilde{x}^j}\bigg\rvert_p. \]
The components of a covector thus transform as follows
\[ \omega_i = \pdv{\tilde{x}^j}{x^i}(p)\tilde{\omega}_j \]
Compare this with the transformation law of components tangent vectors,
\[ \tilde{v}^j = \pdv{\tilde{x}^j}{x^i}(p) v^i \]
Thus, tangent covectors are covariant vectors because they transform in the same way as the partial derivatives, while tangent vectors are called contravariant vectors. This is due to the components of the vectors being the quantities of interest in the past, vectors being treated as n-tuples.
Tensors
A map \(F: V_1 \times \ldots \times V_k \to W\) is multilinear if it is linear as a function of each variable when the others are held fixed. The set of all multilinear maps between a particular domain and codomain is a vector space.
Tensor product
Let \(V\) be a vector space, and \(\omega,\eta \in V^*\). Define a function \(\omega \otimes \eta : V \times V \to \mathbb{R}\)
\[ \omega \otimes \eta(v_1,v_2) = \omega(v_1)\eta(v_2), \]
where the product is regular multiplications of real numbers. Generalising to arbitrary multilinear functionals acting on a product of vector spaces, we get the tensor product. Notice its bilinearity and associativity.
#+ATTR_LATEX :options [Basis for multilinear functions]
Let \(V_1,\ldots,V_k\) be real vector spaces of dimensions \(n_1,\ldots,n_k\) respectivley. Let \((E^{(j)}_i)\) be a basis for \(V_j\) and \((\epsilon^i_{(j)})\) be the corresponding basis for \(V^*_j\). Then the set
\[ \mathcal{B} = \left\{ \epsilon^{i_1}_{(1)}\otimes\ldots\otimes\epsilon^{i_k}_{(k)} : 1\leq i_1 \leq n_1,\ldots,1\leq i_k \leq n_k \right\} \]
is a basis for \(L(V_1,\ldots,V_k:\mathbb{R})\), which therefore has dimension equal to \(n_1\ldots n_k\).
Abstract tensor products
A formal linear combination of elements of a set S is a function \(f : S \to \mathbb{R}\) such that \(f(s) = 0\) for all but finitely many elements of S. The free (real) vector space on S, denoted by \(\mathcal{F}(S)\) is the set of all formal linear combinations of elements of S.
Each element \(f \in \mathcaL{F}(S)\) can be written uniquely in the form \(f = \sum^m_{i=1} a_i x_i\), where \(x_1,\ldots,x_m\) are the elements of S for which \(f(x_i) \neq 0\), and \(a_i = f(x_i)\). We identify \(x \in S\) with the function \(\delta_x \in \mathcal{F}(S)\) that takes the value 1 on \(x\) and zero on all other elements of S. Thus \(S\) is a basis for the free vector space.
#+ATTR_LATEX :options [Characteristic Property of the Free Vector Space]
For any set \(S\) and any vector space \(W\), every map \(A: S \rightarrow W\) has a unique extension to a linear map \(\bar{A}: \mathcal{F}(S) \rightarrow W\)
Now let \(\mathcal{R}\) be the subspace of \(\mathcal{F}(V_1\times\ldots\times V_k)\) spanned by all the elements of the forms
\[ \left(v_{1}, \ldots, a v_{i}, \ldots, v_{k}\right)-a\left(v_{1}, \ldots, v_{i}, \ldots, v_{k}\right) \]
\[ \left(v_{1}, \ldots, v_{i}+v_{i}^{\prime}, \ldots, v_{k}\right)-\left(v_{1}, \ldots, v_{i}, \ldots, v_{k}\right)-\left(v_{1}, \ldots, v_{i}^{\prime}, \ldots, v_{k}\right) \]
with \(v_{j}, v_{j}^{\prime} \in V_{j}, i \in\{1, \ldots, k\}\), and \(a \in \mathbb{R}\).
Define the tensor product of the spaces \(V_{1}, \ldots, V_{k}\), denoted by \(V_{1} \otimes \cdots \otimes V_{k}\), to be the following quotient vector space:
\[ V_{1} \otimes \cdots \otimes V_{k}=\mathcal{F}\left(V_{1} \times \cdots \times V_{k}\right) / \mathcal{R}, \]
and let \(\Pi: \mathcal{F}\left(V_{1} \times \cdots \times V_{k}\right) \rightarrow V_{1} \otimes \cdots \otimes V_{k}\) be the natural projection. The equivalence class of an element \(\left(v_{1}, \ldots, v_{k}\right)\) in \(V_{1} \otimes \cdots \otimes V_{k}\) is denoted by
\[ v_{1} \otimes \cdots \otimes v_{k}=\Pi\left(v_{1}, \ldots, v_{k}\right), \]
and is called the (abstract) tensor product of \(v_{1}, \ldots, v_{k}\). It follows from the definition that abstract tensor products satisfy
\begin{aligned}
v_{1} \otimes \cdots \otimes a v_{i} \otimes \cdots \otimes v_{k}= & a\left(v_{1} \otimes \cdots \otimes v_{i} \otimes \cdots \otimes v_{k}\right) \\
v_{1} \otimes \cdots \otimes\left(v_{i}+v_{i}^{\prime}\right) \otimes \cdots \otimes v_{k}=&\left(v_{1} \otimes \cdots \otimes v_{i} \otimes \cdots \otimes v_{k}\right) \\
&+\left(v_{1} \otimes \cdots \otimes v_{i}^{\prime} \otimes \cdots \otimes v_{k}\right)
\end{aligned}
Note not every element of the tensor product space is of the form \(v_1 \otimes \ldots \otimes v_k\).
An analogue of the basis proposition for multilinear functions holds for abstract tensor product spaces.