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Section A.1 Differential Geometry in Brief

We present here a very brief summary of the basic building blocks of differential geometry.

A differentiable manifold \(M\) is a toplogical space with smooth local coordinates. A coordinate chart \(x=(x^\alpha)\) on an open subset \(U\subset M\) is a map

\begin{equation} x:U\longrightarrow\RR^n\tag{A.1.1} \end{equation}

that is a topological homeomorphism (takes open sets to open sets); \(n\) is the dimension of \(M\text{,}\) and \(\alpha=1,...,n\text{.}\) We will rarely need to worry about the need to use overlapping coordinate charts to cover the entire manifold, but all such overlaps will be assumed to be smooth (\(C^\infty\)) as maps from \(\RR^n\) to itself. A function \(f\) on \(M\) is a map

\begin{equation} f:M\longrightarrow\RR\tag{A.1.2} \end{equation}

which will normally be assumed to be smooth in the sense that \(f\circ x^{-1}\) is smooth as a map from \(\RR^n\) to \(\RR\text{.}\) This setup is illustrated in Figure A.1.1.

Figure A.1.1. Manifolds, coordinates, functions, and vectors.

Also shown in Figure A.1.1 is a vector \(X\) at a point \(P\text{,}\) sometimes written as \(X_P\text{.}\) Vectors are derivative operators; they act on functions by taking a directional derivative at the given point. A vector field, also written as \(X\text{,}\) is then a vector \(X_P\) at every point \(P\in M\text{.}\) More formally,

\begin{equation} X(f) = X^\alpha\frac{\partial}{\partial x^\alpha} \left( f\circ x^{-1} \right)\tag{A.1.3} \end{equation}

for some functions \(X^\alpha\text{,}\) the components of \(X\) in the coordinates \(\{x^\alpha\}\text{,}\) and where there is an implicit sum over \(\alpha\text{.}\) This derivative is often written simply as \(X^\alpha\frac{\partial f}{\partial x^\alpha}\text{.}\) A vector field is smooth if its components are smooth functions.

Consider the sphere \(\SS^2\subset\RR^3\) defined by the equation

\begin{equation} x^2+y^2+z^2=a^2\tag{A.1.4} \end{equation}

where \(a\) is a constant, and which can be described parametrically as

\begin{align} x \amp= a\,\sin\theta\cos\phi ,\tag{A.1.5}\\ y \amp= a\,\sin\theta\sin\phi ,\tag{A.1.6}\\ x \amp= a\,\cos\theta .\tag{A.1.7} \end{align}

Away from the poles, we can use the coordinates \(x^1=\theta\text{,}\) \(x^2=\phi\) (with suitable domains) to map the sphere into \(\RR^2\text{.}\) Any vector field \(X\) on \(\SS^2\) can then be written (away from the poles) as

\begin{equation} X = X^\alpha \frac{\partial}{\partial x^\alpha} = X^\theta \frac{\partial}{\partial x^\theta} + X^\phi \frac{\partial}{\partial x^\phi} .\tag{A.1.8} \end{equation}

This intrinsic description of the sphere makes no explicit use of the rectangular coordinates used in its construction. We can instead give an extrinsic description of the sphere that has the flavor of vector calculus.

Writing \(x\text{,}\) \(y\text{,}\) \(z\) as functions of \(\theta\) and \(\phi\text{,}\) we can express the sphere embedded in \(\RR^3\) in vector parametric form as

\begin{equation} \rr = \rr(\theta,\phi) = x\,\xhat+y\,\yhat+z\,\zhat .\tag{A.1.9} \end{equation}

A basis for the tangent vectors to the sphere at any point (away from the poles) can then be obtained by differentiating \(\rr\text{;}\) the general tangent vector has the form

\begin{equation} \vv = v^\theta \frac{\partial\rr}{\partial\theta} + v^\phi \frac{\partial\rr}{\partial\phi} .\tag{A.1.10} \end{equation}

If we now define

\begin{equation} \vv(f) = \vv\cdot\grad f\tag{A.1.11} \end{equation}

then, for instance,

\begin{equation} \xhat(f) = \xhat\cdot\grad f = \frac{\partial f}{\partial x}\tag{A.1.12} \end{equation}

that is, this action of vectors on functions is precisely the directional derivative. Noting further that \(\xhat=\frac{\partial\rr}{\partial x}\text{,}\) we have related the intuitive notion of vector (“\(x\)-direction”) to a derivative of the position vector \(\rr\) (“\(x\)-derivative of position”) to a derivative operator (“\(x\)-derivative operator”). These three descriptions are equivalent representations of tangent vectors on manifolds, in this case \(\RR^3\text{.}\)

Returning to the sphere, if we now compute, say,

\begin{equation} \frac{\partial\rr}{\partial\phi} = \frac{\partial x}{\partial\phi}\,\xhat + \frac{\partial y}{\partial\phi}\,\yhat\tag{A.1.13} \end{equation}

then

\begin{equation} \frac{\partial\rr}{\partial\phi} (f) = \frac{\partial x}{\partial\phi}\,\frac{\partial f}{\partial x} + \frac{\partial y}{\partial\phi}\,\frac{\partial f}{\partial y} = \frac{\partial f}{\partial\phi}\tag{A.1.14} \end{equation}

and we have shown that (A.1.10) is equivalent to (A.1.8).

A map \(F\) between two manifolds \(M\) and \(N\) not only maps points from one manifold to the other, but also vectors. This situation is shown in Figure A.1.3, where \(y\) represents a coordinate chart on \(N\text{.}\) It is not necessary for \(M\) and \(N\) to have the same dimension.

Figure A.1.3. Manifolds, coordinates, functions, and vectors.

Given a function \(g\) on \(N\text{,}\) we automatically get a function \(g\circ F\) on \(M\text{.}\) Given a vector \(X\) at \(P\in M\text{,}\) we can therefore define a vector \(F_*X\) at \(F(P)\in N\) to be the result of acting on \(g\circ F\) with \(X\text{.}\) Explicitly,

\begin{align} F_*X(g) \amp= X(g\circ F) = X^\alpha \frac{\partial}{\partial x^\alpha} \left(g\circ F\circ x^{-1}\right)\notag\\ \amp= X^\alpha \frac{\partial}{\partial y^\beta} \left(g\circ y^{-1}\right) \frac{\partial}{\partial x^\alpha} \left(y^\beta\circ F\circ x^{-1}\right)\tag{A.1.15} \end{align}

where the last term is the Jacobian matrix of \(F\text{,}\) often denoted simply \(\frac{\partial y^\beta}{\partial x^\alpha}\text{.}\) Comparing (A.1.15) with (A.1.3) leads to

\begin{equation} (F_*X)^\beta = \frac{\partial y^\beta}{\partial x^\alpha} X^\alpha\tag{A.1.16} \end{equation}

which relates the components of \(F_*X\) to those of \(X\text{.}\) Thus, the components of \(F_*X\) are given by multiplying the components of \(X\) by the Jacobian matrix.

Consider the case where \(F:\SS^2\longrightarrow\SS^2\) is a rotation of the sphere about the \(z\)-axis by an angle \(\psi\text{.}\) This transformation is given by

\begin{equation} (\theta,\phi) \longmapsto (\theta,\phi+\psi)\tag{A.1.17} \end{equation}

and it is straightforward to show that the Jacobian matrix of \(F\) is the \(2\times2\) identity matrix. Thus, \(F_*\) takes a vector of the form (A.1.8) at a given point \(P\in\SS^2\) to a vector with the same components at \(F(P)\text{.}\)

However, if \(F\) were a rotation about some other axis, it is not obvious what it does to the coordinates \((\theta,\phi)\text{,}\) and therefore not obvious what the Jacobian matrix is.

Switching from the intrinsic description to the extrinsic description, any rotation can be expressed as a \(3\times3\) matrix \(M\) acting on the position vector, thought of as the column vector

\begin{equation} w = \begin{pmatrix}x\\y\\z\end{pmatrix} ,\tag{A.1.18} \end{equation}

that is, any rotation can be described as

\begin{equation} w \longmapsto Mw .\tag{A.1.19} \end{equation}

Since \(M\) is a constant, it pulls through the computation of the Jacobian matrix! Thus, if we expand a tangent vector \(\vv\) in the form (A.1.10) as a vector in \(\RR^3\), so that

\begin{equation} \vv = v^x\,\xhat + v^y\,\yhat + v^z\,\zhat\tag{A.1.20} \end{equation}

again thought of as a column vector, namely

\begin{equation} v = \begin{pmatrix}v^x\\v^y\\v^z\end{pmatrix} ,\tag{A.1.21} \end{equation}

then it is straightforward to show that

\begin{equation} F_*v = Mv\tag{A.1.22} \end{equation}

which can be converted to either of the forms (A.1.10) or (A.1.8).

This example shows one of the advantages of working in the extrinsic description. This approach underlies much of the computation elsewhere in this book, with \(\RR^3\) replaced by \(\RR^{n\times n}\text{,}\) the space of \(n\times n\) matrices, and \(\SS^2\) replaced by a matrix Lie group.