1 Fundamental notions

We start by introducing some notions that are fundamental for the study of curved spaces. Doing so will lead to a deeper understanding of some concepts from Linear Algebra and Analysis.

1.1 Points, vectors and the tangent space

Recall that we define \(\mathbb{R}^n\) as ordered \(n\)-tuples \(p=(x_1,\ldots,x_n)\) of scalars \(x_i \in \mathbb{R},\) \(1\leqslant i\leqslant n.\) We also consider column vectors of length \(n\) with real entries \[\vec{v}=\begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}.\] We write \(M_{m,n}(\mathbb{R})\) for the set of \((m\times n)\)-matrices with real entries. A column vector of length \(n\) may be thought of as an \((n\times 1)\)-matrix, hence we write \(M_{n,1}(\mathbb{R})\) for the set of such column vectors. Clearly we have a bijective map \[\Psi_n : \mathbb{R}^n \to M_{n,1}(\mathbb{R}), \quad (x_1,\ldots,x_n) \mapsto \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}\] which writes the entries of an \(n\)-tuple into a column vector. Because of this map, we may avoid a distinction between \(\mathbb{R}^n\) and \(M_{n,1}(\mathbb{R})\) and pretend they are the same thing. This was done so in Linear Algebra. In geometry, it turns out to be useful to think of \(\mathbb{R}^n\) and \(M_{n,1}(\mathbb{R})\) as different sets. The elements of \(\mathbb{R}^n\) are interpreted as points and will be denoted by \(p,q,r\ldots.\) The elements of \(M_{n,1}(\mathbb{R})\) are interpreted as vectors in \(\mathbb{R}^n\) that are attached to the origin \(0_{\mathbb{R}^n}=(0,0,\ldots,0)\in \mathbb{R}^n.\) They will be denoted by \(\vec{u},\vec{v},\vec{w},\ldots .\)

Already in elementary geometry the situation occurs where we consider vectors in \(\mathbb{R}^n\) that are not attached to the origin \(0_{\mathbb{R}^n},\) but rather to some other point \(p \in \mathbb{R}^n.\) Think for instance of the normal vector of a plane in \(\mathbb{R}^3\) not containing the origin \(0_{\mathbb{R}^3}.\)

In order to deal with vectors that are not attached to the origin, but to a point \(p \in \mathbb{R}^n,\) we introduce the so-called tangent space of \(\mathbb{R}^n\) at \(p,\) \[T_{p}\mathbb{R}^n=\left\{\vec{v}_{p}\,|\, \vec{v} \in M_{n,1}(\mathbb{R})\right\}.\] The element \(\vec{v}_{p} \in T_p\mathbb{R}^n\) is to be interpreted as attaching the vector \(\vec{v} \in M_{n,1}(\mathbb{R})\) at the basepoint \(p \in \mathbb{R}^n.\) The elements of \(T_p\mathbb{R}^n\) are called tangent vectors with basepoint \(p\). Observe that for all \(p \in \mathbb{R}^n\) the tangent space \(T_p\mathbb{R}^n\) is a vector space over \(\mathbb{R}\) when equipped with vector addition \(+_{T_p\mathbb{R}^n} : {T_p\mathbb{R}^n} \times {T_p\mathbb{R}^n} \to {T_p\mathbb{R}^n}\) defined by the rule \[\vec{v}_{p}+_{T_p\mathbb{R}^n}\vec{w}_{p}=(\vec{v}+_{M_{n,1}(\mathbb{R})}\vec{w})_{p}\] for all \(\vec{v}_{p},\)\(\vec{w}_{p} \in T_p\mathbb{R}^n\) and scalar multiplication \(\cdot_{T_p\mathbb{R}^n} : \mathbb{R}\times {T_p\mathbb{R}^n} \to {T_p\mathbb{R}^n}\) defined by the rule \[s\cdot_{T_p\mathbb{R}^n}\vec{v}_{p}=(s\cdot_{M_{n,1}(\mathbb{R})}\vec{v})_{p}\] for all \(s \in \mathbb{R}\) and all \(\vec{v}_{p} \in T_p\mathbb{R}^n.\) Here \(+_{M_{n,1}(\mathbb{R})}\) denotes usual component-wise addition of column vectors and \(\cdot_{M_{n,1}(\mathbb{R})}\) denotes usual component-wise scalar multiplication of a column vector by a scalar. Clearly, for all \(p \in \mathbb{R}^n\) we have a vector space isomorphism \[T_p\mathbb{R}^n \to M_{n,1}(\mathbb{R}), \quad \vec{v}_{p} \mapsto \vec{v}\] which simply “forgets“ the basepoint \(p \in \mathbb{R}^n.\) We can thus think of \(T_p\mathbb{R}^n\) as a copy of \(M_{n,1}(\mathbb{R})\) attached to \(p \in \mathbb{R}^n.\) The union of all these copies of \(\mathbb{R}^n\) is known as the tangent bundle of \(\mathbb{R}^n\) \[T\mathbb{R}^n=\bigcup_{p \in \mathbb{R}^n}T_p\mathbb{R}^n=\bigcup_{p \in \mathbb{R}^n}\left\{\vec{v}_{p}\,|\, \vec{v} \in M_{n,1}(\mathbb{R})\right\}.\] At this point the name tangent space is a bit confusing, since it is unclear to what \(T_p\mathbb{R}^n\) is tangent to. This will be clarified later on. If \(U\subset \mathbb{R}^n\) is an open subset, we define likewise \[TU=\bigcup_{p \in U}T_p\mathbb{R}^n.\] Observe that for each \(p \in \mathbb{R}^n\) the tangent space \(T_p\mathbb{R}^n\) is equipped with an ordered basis \[\mathbf{e}_p^{(n)}=\big((\vec{e}_1)_{p},\ldots,(\vec{e}_n)_{p}\big),\] where \(\{\vec{e}_1,\ldots,\vec{e}_n\}\) denotes the standard basis of \(M_{n,1}(\mathbb{R}).\) For all \(p \in \mathbb{R}^n\) we call \(\mathbf{e}^{(n)}_p\) the ordered standard basis of \(T_p\mathbb{R}^n.\)

Whenever \(n\) is clear from the context we simply write \(\mathbf{e}_p\) instead of \(\mathbf{e}^{(n)}_p.\)

Recall that a pair \((V,\langle\cdot{,}\cdot\rangle)\) consisting of a vector space \(V\) over \(\mathbb{R}\) and an inner product \(\langle\cdot{,}\cdot\rangle: V\times V \to \mathbb{R}\) is called a Euclidean space.¹

An inner product on an vector \(V\) over \(\mathbb{R}\) is a positive definite symmetric bilinear form \(\langle\cdot{,}\cdot\rangle: V\times V \to \mathbb{R}.\)

We can turn each tangent space into a Euclidean space:

We will henceforth always assume that \(T_p\mathbb{R}^n\) is equipped with \(\langle\cdot{,}\cdot\rangle_p.\) Whenever no confusion can arise about the point \(p\) at which \(\langle\cdot{,}\cdot\rangle_p\) is computed, we will usually simply write \(\langle\cdot{,}\cdot\rangle.\)

Occasionally it is useful to turn a tangent vector \(\vec{v}_p \in T_p\mathbb{R}^n\) into a point \(q \in \mathbb{R}^n.\) This is done by mapping a tangent vector \(\vec{v}_p \in T_p\mathbb{R}^n\) to its “endpoint”. More precisely, we define:

1.2 Smooth maps, diffeomorphisms and the differential

We recall some facts from Analysis II, but now with a slightly more geometric perspective.

For \(n \in \mathbb{N}\) we let \(\{e_1,\ldots,e_n\}\) – here interpreted as points – denote the standard basis of \(\mathbb{R}^n,\) that is \(e_1=(1,0,0,\ldots,0),\) \(e_2=(0,1,0,\ldots,0)\) and so on. Let \(U\subset\mathbb{R}^n\) be an open set and consider a map \(f : U \to \mathbb{R}^m.\) Recall that for all \(p \in U\) and \(1\leqslant i\leqslant n\) we define the partial derivative of \(f\) in the coordinate direction \(i\) as \[\partial_if(p)=\lim_{h \to 0} \frac{1}{h}\left(f(p+he_i)-f(p)\right),\] provided the limit exists. Recall also from Analysis II that the map \(f : U \to \mathbb{R}^m\) is continuously differentiable if and only if²

This is a theorem, not a definition!

for all \(1\leqslant i\leqslant n\)

1. the partial derivative \(\partial_if(p)\) exists for all \(p \in U\);

2. the map \(\partial_i f : U \to \mathbb{R}^m,\) \(p \mapsto \partial_if(p)\) is continuous.

Recursively, we can define higher derivatives. For \(k \in \mathbb{N}, k \geqslant 2\) we call \(f : U \to \mathbb{R}^m\) \(k\)-times continuously differentiable if \(\partial_i f : U \to \mathbb{R}^m\) is \((k-1)\)-times continuously differentiable for all \(1\leqslant i\leqslant n.\) We write \[C^k(U,\mathbb{R}^m)=\left\{f : U \to \mathbb{R}^m\, |\, f \text{ is }k\text{-times continuously differentiable}\right\}\] and

Throughout this module we will almost exclusively consider smooth maps.

Diffeomorphisms are smooths maps that are bijective and admit a smooth inverse:

Recall from Analysis that if \(f : U \to V\) is a diffeomorphism, then \(n=m\) and moreover, for all \(p \in U\) the linear map \(f_*|_{p} : T_p\mathbb{R}^n \to T_{f(p)}\mathbb{R}^n\) is invertible.

If \(f : U \to \mathbb{R}^m\) is a smooth and injective map, we say \(f\) is a diffeomorphism onto its image, provided the inverse map \(f^{-1} : \operatorname{Im}(f) \to U\) is smooth as well. Here as usual we define \[\operatorname{Im}(f)=f(U)=\{q \in \mathbb{R}^m | q=f(p),\, p \in U\}.\]

From the chain rule in Analysis II we conclude:

1.3 Vector fields and the gradient

Recall that for all \(p \in \mathbb{R}^n\) the tangent space \(T_p\mathbb{R}^n\) is equipped with a basis given by attaching the standard basis \(\{\vec{e}_1,\ldots,\vec{e}_n\}\) of \(M_{n,1}(\mathbb{R})\) at \(p.\) We may think of attaching the \(i\)-th standard basis vector at \(p\) as a map from \(\mathbb{R}^n\) to \(T\mathbb{R}^n.\) That is, we define \[\frac{\partial}{\partial x_i} : \mathbb{R}^n \to T\mathbb{R}^n, \qquad p \mapsto \frac{\partial}{\partial x_i}(p)=\left(\vec{e}_i\right)_p.\] The mappings \(\frac{\partial}{\partial x_i}\) are examples of vector fields:

A vector field simply attaches a tangent vector \(v_p\) to every point \(p\) of its domain of definition. Vector fields appear naturally in physics. For instance, an electromagnetic field is an example of a vector field. Likewise, in the classical Newtonian theory of gravity, the gravitational field is an example of a vector field.

Every smooth function gives rise to a vector field:

1.4 The cotangent space and the exterior derivative

Recall from Linear Algebra II that if \(V\) is a vector space over \(\mathbb{R},\) then its dual vector space \(V^*\) consists of the linear maps \(f : V \to \mathbb{R}\) with vector addition defined by the rule \[(f+_{V^*}g)(v)=f(v)+_{\mathbb{R}}g(v)\] for all \(f,g \in V^*\) and all \(v \in V\) and scalar multiplication defined by the rule \[(s\cdot_{V^*}f)=s\cdot_{\mathbb{R}}f(v)\] for all \(f \in V^*,\) all \(s \in \mathbb{R}\) and all \(v \in V.\)

We can think of row vectors of length \(n\) with real entries, that is the elements of \(M_{1,n}(\mathbb{R}),\) as elements of the dual vector space of the vector space \(M_{n,1}(\mathbb{R})\) of column vectors of length \(n\) with real entries. This is done by interpreting \(\vec{\nu} \in M_{1,n}(\mathbb{R})\) as a linear map \(M_{n,1}(\mathbb{R}) \to \mathbb{R}\) given by \[\vec{\nu}(\vec{v})=\vec{\nu}\vec{v} \in \mathbb{R},\] where on the right hand side we use matrix multiplication of the row vector \(\vec{\nu}\) and the column vector \(\vec{v}.\) In doing so, the standard basis of \(M_{1,n}(\mathbb{R})\) given by \(\{\vec{\varepsilon}_1,\ldots,\vec{\varepsilon}_n\}\) can be interpreted as as basis of \((M_{n,1}(\mathbb{R}))^*.\) Here \(\vec{\varepsilon}_i\) denotes the row vector of length \(n\) whose \(i\)-th entry is \(1\) with all other entries \(0.\)

Let \(n \in \mathbb{N}\) and \(p \in \mathbb{R}^n.\) The dual vector space of the tangent space \(T_p\mathbb{R}^n\) is called the cotangent space at \(p \in M\) and denoted by \(T^*_p\mathbb{R}^n:=(T_p\mathbb{R}^n)^*.\) We write an element of \(T^*_p\mathbb{R}^n\) as \(\vec{\nu}_p\) with \(p \in \mathbb{R}^n\) and \(\vec{\nu} \in M_{1,n}(\mathbb{R}).\) Hence we have \[T^*_p\mathbb{R}^n=\left\{\vec{\nu}_p\,|\, \vec{\nu}\in M_{1,n}(\mathbb{R})\right\}.\] The elements of \(T^*_p\mathbb{R}^n\) are called cotangent vectors with basepoint \(p\). The union of all cotangent spaces is the so-called cotangent bundle \[T^*\mathbb{R}^n=\bigcup_{p \in \mathbb{R}^n}T^*_p\mathbb{R}^n=\bigcup_{p \in \mathbb{R}^n}\left\{\vec{\nu}_p\,|\, \vec{\nu}\in M_{1,n}(\mathbb{R})\right\}.\] As in the case of the tangent space, each cotangent space \(T_p^*\mathbb{R}^n\) is equipped with an ordered basis \(\boldsymbol{\varepsilon}_{p}^{(n)}=((\vec{\varepsilon}_1)_p,\ldots,(\vec{\varepsilon}_n)_p).\)

Differential Geometry

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