5 Differential Geometry

5.0.1 Forms

Forms are functions that take vectors as input and return scalars. A 1-form takes a single vector as input, a 2-form takes two vectors as input, and so on.

definition (1-form = linear functional = covector = covariant vector) A linear, real-valued function of a single vector input, \(\boldsymbol{\omega}: \mathbf{V} \rightarrow \mathbb{R}\), which satisfies the following conditions for all vectors and scalars:

(Additivity) \(\boldsymbol{\omega}(\mathbf{u} + \mathbf{v}) = \boldsymbol{\omega}(\mathbf{u}) + \boldsymbol{\omega}(\mathbf{v})\).
(Homogeneity) \(\boldsymbol{\omega}(k\mathbf{v}) = k \cdot \boldsymbol{\omega}(\mathbf{v})\).

where - \(\mathbf{V}\) is a real vector space. - \(\mathbb{R}\) is the set of real numbers. - \(\boldsymbol{\omega}\) is a 1-form. - \(\mathbf{u}, \mathbf{v}\) are vectors in \(\mathbf{V}\). - \(k\) is a scalar.

definition (2-form) A bilinear, antisymmetric, real-valued function of two vector inputs, \(\boldsymbol{\omega}: \mathbf{V} \times \mathbf{V} \rightarrow \mathbb{R}\), which satisfies the following conditions for all vectors and scalars:

(Bilinearity) \(\boldsymbol{\omega}(a\mathbf{u} + b\mathbf{u}', \mathbf{v}) = a \cdot \boldsymbol{\omega}(\mathbf{u}, \mathbf{v}) + b \cdot \boldsymbol{\omega}(\mathbf{u}', \mathbf{v})\) and \(\boldsymbol{\omega}(\mathbf{u}, a\mathbf{v} + b\mathbf{v}') = a \cdot \boldsymbol{\omega}(\mathbf{u}, \mathbf{v}) + b \cdot \boldsymbol{\omega}(\mathbf{u}, \mathbf{v}')\).
(Antisymmetry) \(\boldsymbol{\omega}(\mathbf{u}, \mathbf{v}) = -\boldsymbol{\omega}(\mathbf{v}, \mathbf{u})\).

where - \(\mathbf{V}\) is a real vector space. - \(\mathbb{R}\) is the set of real numbers. - \(\boldsymbol{\omega}\) is a 2-form. - \(\mathbf{u}, \mathbf{u}', \mathbf{v}, \mathbf{v}'\) are vectors in \(\mathbf{V}\). - \(a, b\) are scalars.

definition (k-form) A multilinear, completely antisymmetric, real-valued function of \(k\) vector inputs, \(\boldsymbol{\omega}: \mathbf{V}^k \rightarrow \mathbb{R}\), which satisfies the following conditions for all vectors and scalars:

(k-Linearity) \(\boldsymbol{\omega}(\mathbf{v}_1, \dots, a\mathbf{v}_i + b\mathbf{v}_i', \dots, \mathbf{v}_k) = a \cdot \boldsymbol{\omega}(\mathbf{v}_1, \dots, \mathbf{v}_i, \dots, \mathbf{v}_k) + b \cdot \boldsymbol{\omega}(\mathbf{v}_1, \dots, \mathbf{v}_i', \dots, \mathbf{v}_k)\) for each argument \(i\) where \(1 \leq i \leq k\).
(Complete Antisymmetry) Swapping any two vector inputs reverses the sign of the output: \(\boldsymbol{\omega}(\dots, \mathbf{u}, \dots, \mathbf{v}, \dots) = -\boldsymbol{\omega}(\dots, \mathbf{v}, \dots, \mathbf{u}, \dots)\).

where - \(\mathbf{V}\) is a real vector space. - \(\mathbf{V}^k\) is the \(k\)-fold Cartesian product of \(\mathbf{V}\). - \(\mathbb{R}\) is the set of real numbers. - \(\boldsymbol{\omega}\) is a \(k\)-form. - \(\mathbf{v}_1, \dots, \mathbf{v}_k, \mathbf{v}_i', \mathbf{u}, \mathbf{v}\) are vectors in \(\mathbf{V}\). - \(k\) is a positive integer representing the degree of the form. - \(a, b\) are scalars.

5.0.2 Tensors

definition (Multilinear Function) A function from the \(k\)-fold product of a vector space to the real numbers, \(T: V^k \rightarrow \mathbb{R}\), where the following conditions apply for each argument \(i\) (\(1 \leq i \leq k\)):

\(T(v_1, \dots, v_i + v_i', \dots, v_k) = T(v_1, \dots, v_i, \dots, v_k) + T(v_1, \dots, v_i', \dots, v_k)\).
\(T(v_1, \dots, av_i, \dots, v_k) = a \cdot T(v_1, \dots, v_i, \dots, v_k)\).

where - \(V\) is a vector space over \(\mathbb{R}\). - \(V^k\) is the \(k\)-fold Cartesian product \(V \times \dots \times V\). - \(k\) is a positive integer representing the number of arguments, also called the degree of the function. - \(v_1, \dots, v_k, v_i' \in V\) are vectors. - \(a \in \mathbb{R}\) is a scalar.

definition (k-Tensor) A multilinear function from the \(k\)-fold product of a vector space to the real numbers, \(T: V^k \rightarrow \mathbb{R}\), where the following condition applies:

The function \(T\) is linear in each of its \(k\) arguments.

where - \(V\) is a vector space. - \(V^k\) is the \(k\)-fold Cartesian product \(V \times \dots \times V\). - \(k\) is a positive integer representing the degree of the tensor. - \(\mathbb{R}\) is the set of real numbers. - \(\mathcal{L}^k(V)\) (or \(\mathcal{J}^k(V)\)) is the vector space of all \(k\)-tensors on \(V\).

\(T(v_1, \dots, v_i + v_i', \dots, v_k) = T(v_1, \dots, v_i, \dots, v_k) + T(v_1, \dots, v_i', \dots, v_k)\)
\(T(v_1, \dots, av_i, \dots, v_k) = a \cdot T(v_1, \dots, v_i, \dots, v_k)\)

definition (k-Fold Product) A set constructed by the Cartesian product of \(k\) copies of a single set, \(X^k\), where the following condition applies:

The resulting set consists of all ordered \(k\)-tuples where each coordinate is an element of the original set.

where - \(k\) is a positive integer representing the number of repetitions. - \(X\) is the underlying set, topological space, or vector space. - \(X^k\) (or \(V^k\)) denotes the \(k\)-fold Cartesian product \(X \times \dots \times X\).

definition (Cartesian Product) A set consisting of all possible ordered combinations of elements from a collection of sets, denoted by \(X \times Y\) for two sets, \(X_1 \times \dots \times X_n\) for \(n\) sets, or \(\prod_{\alpha \in A} X_\alpha\) for an indexed family, where the following conditions apply:

For two sets, the product contains all ordered pairs \((x, y)\) where the first component is from the first set and the second component is from the second.
For a finite collection, the product consists of all ordered \(n\)-tuples \((x_1, \dots, x_n)\) where each \(x_i\) is an element of the corresponding factor \(X_i\).
For an arbitrary indexed family, the product is the set of all functions \(x\) from the index set \(A\) to the union of the sets such that \(x(\alpha) \in X_\alpha\) for each \(\alpha \in A\).

where - \(X, Y, X_i, X_\alpha\) are non-empty sets - \((x, y)\) is an ordered pair - \((x_1, \dots, x_n)\) is an ordered \(n\)-tuple - \(A\) (or \(\Lambda\)) is the index set used to label the family of sets

5.0.3 Covariant, Contravariant, and Coordinate Systems

The context if this section is the following. In physics and mathematics, points exist independently of the coordinates used to represent them.

First, we begin a discussion on covariant and contravariant vectors.

Covariant vectors are linear functionals.

definition (Linear Functional) A linear transformation from a vector space \(V\) to its associated field of scalars \(K\), The mapping \(f: V \rightarrow K\) must satisfy the following linearity condition:

\(f(ax + by) = af(x) + bf(y)\)

where - \(x, y \in V\) are vectors - \(a, b \in K\) are scalars - \(V\) is a vector space - \(K\) is the field of scalars.

definition (k-Covector) An alternating \(k\)-linear function from the \(k\)-fold product of a vector space to the real numbers, \(T: V^k \rightarrow \mathbb{R}\), where the following conditions apply:

The function \(T\) is linear in each of its \(k\) arguments.
\(T(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (\text{sgn } \sigma) T(v_1, \dots, v_k)\) for every permutation \(\sigma \in S_k\).

where - \(V\) is a vector space. - \(k\) is a positive integer representing the degree of the covector. - \(v_1, \dots, v_k \in V\) are vectors. - \(A^k(V)\) (or \(\Lambda^k(V^*)\)) is the vector space of all \(k\)-covectors on \(V\). - \(\text{sgn } \sigma\) is the sign of the permutation. - A \(k\)-covector is also called a multicovector of degree k or an alternating k-tensor.

definition (k-Covector Field) A function assigning a \(k\)-covector to each point of a manifold, \(\omega: M \rightarrow \Lambda^k(T^*M)\), where the following condition applies:

For each point \(p\) in \(M\), \(\omega(p)\) is an alternating \(k\)-linear function on the tangent space \(T_pM\).

where - \(M\) is a smooth manifold. - \(p\) is a point in \(M\). - \(T_pM\) is the tangent space of \(M\) at \(p\). - \(T^*_pM\) is the cotangent space of \(M\) at \(p\). - \(\Lambda^k(T^*_pM)\) (or \(A^k(T_pM)\)) is the space of all alternating \(k\)-tensors on the tangent space \(T_pM\). - \(\Lambda^k(T^*M)\) is the \(k\)-th exterior power of the cotangent bundle. - A \(k\)-covector field is also called a differential \(k\)-form.

example (k-Covector) Let

\[ V = \mathbb{R}^2. \]

Define the function

\[ T : V^2 \rightarrow \mathbb{R} \]

\[ T((a,b),(c,d)) = ad-bc. \]

where \[a,b,c,d \in \mathbb{R}\].

Let

\[ v_1=(1,2), \qquad v_2=(3,4). \]

Then

\[ T(v_1,v_2) = (1)(4)-(2)(3) = -2. \]

Interchanging the vectors gives

\[ T(v_2,v_1) = (3)(2)-(4)(1) = 2 = -T(v_1,v_2). \]

Therefore, \(T\) changes sign when its arguments are exchanged, so \(T\) is alternating.

Furthermore, \(T\) is linear in each argument. Hence, \(T\) is an alternating \(2\)-linear function.

Therefore, \(T\) is a \(2\)-covector on \(\mathbb{R}^2\).

5.0.4 Functions

definition (Alternating Function) A property of a \(k\)-linear function from the \(k\)-fold product of a vector space to the real numbers, \(T: V^k \rightarrow \mathbb{R}\), where the following conditions apply:

\(T(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (\text{sgn } \sigma) T(v_1, \dots, v_k)\) for every permutation \(\sigma \in S_k\).
\(T(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -T(v_1, \dots, v_j, \dots, v_i, \dots, v_k)\) for any interchange of two arguments.
\(T(v_1, \dots, v_k) = 0\) whenever two of the vectors \(v_1, \dots, v_k\) are equal.

where - \(V\) is a vector space. - \(V^k\) is the \(k\)-fold Cartesian product \(V \times \dots \times V\). - \(S_k\) is the permutation group of \(k\) objects. - \(\text{sgn } \sigma\) is the sign of the permutation \(\sigma\), which is \(+1\) if the permutation is even and \(-1\) if it is odd. - \(v_1, \dots, v_k\) are vectors in \(V\). - \(k\) is a positive integer representing the degree of the function.

definition (Alternating k-linear function) A multilinear function from the \(k\)-fold product of a vector space to the real numbers, \(f: V^k \rightarrow \mathbb{R}\), where the following conditions apply:

\(f(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (\text{sgn } \sigma) f(v_1, \dots, v_k)\) for all \(\sigma \in S_k\).
\(f(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -f(v_1, \dots, v_j, \dots, v_i, \dots, v_k)\) for any interchange of two arguments.
\(f(v_1, \dots, v_k) = 0\) whenever two of the vectors \(v_1, \dots, v_k\) are equal.

where - \(V\) is a vector space. - \(V^k\) is the \(k\)-fold Cartesian product \(V \times \dots \times V\). - \(k\) is a positive integer representing the degree of the function. - \(S_k\) is the permutation group of \(k\) objects. - \(\text{sgn } \sigma\) is the sign of the permutation \(\sigma\). - \(v_1, \dots, v_k \in V\) are vectors. - \(A^k(V)\) (or \(\Lambda^k(V^*)\)) is the vector space of all alternating \(k\)-linear functions on \(V\).

k-covector, multicovector of degree k, and alternating k-tensor are synonyms for an alternating \(k\)-linear function.

5.0.5 Differential Form

definition (Differential k-Form) A function assigning an alternating k-linear function to each point of a manifold \(M\), where the following condition applies:

For each point \(p \in M\), \(\omega(p)\) is a \(k\)-covector on the tangent space \(T_pM\).

where - \(M\) is a smooth manifold. - \(k\) is a non-negative integer representing the degree of the form. - \(T_pM\) is the tangent space to \(M\) at \(p\). - \(T^*_p M\) (or \(T^*_p(M)\)) is the cotangent space at \(p\), defined as the dual space of the tangent space \(T_pM\). - \(\Lambda^k(T^*_p M)\) (or \(A^k(T_pM)\)) is the vector space of all alternating \(k\)-tensors (also called \(k\)-covectors or multicovectors) on \(T_pM\). - \(\omega\) is a smooth section of the vector bundle \(\Lambda^k(T^*M)\), the \(k\)-th exterior power of the cotangent bundle.

5.0.6 Appendix

definition (Linear Function) A function from a vector space, \(f: \mathbf{V} \rightarrow \mathbb{R}\), that satisfies the following conditions for all vectors and scalars:

(Additivity) \(f(\mathbf{u} + \mathbf{v}) = f(\mathbf{u}) + f(\mathbf{v})\) .
(Homogeneity) \(f(k\mathbf{v}) = k \cdot f(\mathbf{v})\).

where - \(\mathbf{V}\) is a real vector space. - \(\mathbf{u}, \mathbf{v} \in \mathbf{V}\). - \(k\) is a scalar.

4 Topology