4 Intrinsic surface geometry
An important observation in geometry is that the Gauss curvature of an embedded surface can be expressed in terms of the first fundamental form only. Gauss, who discovered this fact, was so astonished by it that he called it a “Theorema Egregium” (Latin for “Remarkable Theorem”). Geometric quantities associated to a surface which can be obtained from computing inner products between tangent vectors – that is, quantities that are computable once we know the first fundamental form – are called intrinsic. Intrinsic quantities are in contrast to extrinsic quantities which cannot be computed from knowing the first fundamental form alone. Prototypical examples of extrinsic quantities associated to an embedded surface \(M\subset \mathbb{R}^3\) are the second fundamental form, the mean curvature and the unit normal field. The intuition for intrinsic vs extrinsic is that intrinsic quantities do not rely on the ambient space \(\mathbb{R}^3\) in which the surface is embedded, whereas extrinsic quantities do.
Recall that the Gauss curvature is the product of the principal curvatures which can be computed as the signed curvature of the curve cut out of the surface by intersecting it with a suitable affine 2-plane. At \(p \in M\) the affine 2-plane is spanned by a unit normal vector \(N(p)\) and a tangent vector \(\vec{v}_p.\) The unit normal vector \(N(p)\) being an extrinsic quantity, it is not clear at all that the Gauss curvature can be expressed without involving \(N(p),\) this is however the case as we will see below. Gauss’ Theorema Egregium lead mathematicians to consider geometric spaces which are not necessarily embedded in a surrounding ambient space such as \(\mathbb{R}^3.\) This point of view is relevant in particular in physics. As far as we know our universe does not sit inside a larger ambient universe, but is a geometric space in itself.
4.1 The Gauss–Codazzi equations
Specifying an immersed surface \(F : U \to \mathbb{R}^3\) involves choosing \(3\) functions \(F_1,F_2,F_3 : U \to \mathbb{R}.\) Recall that to an immersed surface \(F : U \to \mathbb{R}^3\) we associated two maps \(g,A : U \to M_{2,2}(\mathbb{R})\) taking values in the symmetric \(2\times 2\)-matrices. The map \(g\) encodes the first fundamental form and the map \(A\) encodes the second fundamental form. We can change our view point and prescribe two matrix-valued maps \(g,A\) on an open set \(U\subset \mathbb{R}^2\) and ask whether \(g\) and \(A\) arise from an immersion \(F : U \to \mathbb{R}^3\) via the expressions given in (3.8) and (3.11). Thinking of \(g_{ij},A_{ij}\) for \(1\leqslant i,j\leqslant 2\) as given and \(F_1,F_2,F_3\) as unknown functions, the equations (3.8) and (3.11) are a system of partial differential equations. Partial differential equations are the (generally speaking more complicated) counterparts to ordinary differential equations, the key difference being that the sought after functions are allowed to depend on more than one variable. Many important laws of nature can be phrased as partial differential equations, in particular, the so-called Einstein field equations describing gravity, the Schrödinger equation arising in quantum mechanics and the Maxwell equations governing the laws of electromagnetism. Understanding partial differential equations is a fundamental part of modern mathematics.
The two systems (3.8) and (3.11) give us \(6\) equations for \(3\) unknown functions \(F_1,F_2,F_3.\) Roughly speaking, whenever we have more equations (here \(6\)) than unknowns (here \(3\)) we should expect some compatibility conditions among the equations so that we can find any solutions. For historical reasons, in the theory of partial differential equations compatibility conditions are often called integrability conditions.
Proposition 4.5 • Gauss–Codazzi equations ➔
below.Let \(U\subset \mathbb{R}^2\) be an open subset and \(g_{ij},A_{ij} : U \to \mathbb{R}\) smooth functions for \(1\leqslant i,j\leqslant 2\) with \(g_{12}=g_{21}\) and \(A_{12}=A_{21}.\) Then the Gauss – and Codazzi equations \[\tag{4.7} R_{ijkm}=A_{ik}A_{jm}-A_{jk}A_{im} \qquad \text{and} \qquad \partial_jA_{ik}-\partial_iA_{jk}=\Gamma^l_{jk}A_{il}-\Gamma^l_{ik}A_{jl},\] are necessary conditions for the existence of a smooth immersion \(F : U \to \mathbb{R}^3\) whose associated functions via (3.9) and (3.12) are \(g_{ij}\) and \(A_{ij}.\)
Let \(F : U \to \mathbb{R}^3\) be an immersed surface so that \[\mathbf{c}(q):=(\partial_1 F(q),\partial_2 F(q),G(q))\] is an ordered basis of \(M_{3,1}(\mathbb{R})\) for all \(q \in U.\) For all \(q \in U\) and all \(1\leqslant i,j\leqslant 2,\) the vector \(\partial^2_{ij}F(q) \in M_{3,1}(\mathbb{R})\) can thus be written as a linear combination of the elements of \(\mathbf{c}(q).\) We can therefore find unique functions \(\Gamma^k_{ij} : U \to \mathbb{R}\) and \(B_{ij} : U \to \mathbb{R}\) for \(1\leqslant i,j,k\leqslant 2\) so that \[\tag{4.1} \partial^2_{ij}F(q)=\Gamma^1_{ij}(q)\partial_1 F(q)+\Gamma^2_{ij}(q)\partial_2 F(q)+B_{ij}(q)G(q).\] Taking the inner product with \(G\) we obtain \[A_{ij}=G\cdot \partial^2_{ij}F=\Gamma^1_{ij}\left(G\cdot \partial_1 F\right)+\Gamma^2_{ij}\left(G\cdot \partial_2 F\right)+B_{ij}\left(G\cdot G\right)=B_{ij},\] where we suppress the base point, we use (3.12), \(G(q)\cdot G(q)=1\) and that \(G(q)\) is orthogonal to \(\partial_1 F(q)\) and to \(\partial_2 F(q).\) Taking inner products with \(\partial_1 F\) and \(\partial_2 F\) we obtain \[\partial_1 F \cdot \partial^2_{ij}F=\Gamma^1_{ij}\left(\partial_1 F\cdot \partial_1 F\right)+\Gamma^2_{ij}\left(\partial_1 F\cdot \partial_2 F\right)=\Gamma^1_{ij}g_{11}+\Gamma^2_{ij}g_{12}\] and \[\partial_2 F \cdot \partial^2_{ij}F=\Gamma^1_{ij}\left(\partial_2 F\cdot \partial_1 F\right)+\Gamma^2_{ij}\left(\partial_2 F\cdot \partial_2 F\right)=\Gamma^1_{ij}g_{21}+\Gamma^2_{ij}g_{22},\] where we use (3.8).
Remark 4.1 • Einstein Summation convention
In what follows we employ a useful notational convention going back to A. Einstein. Whenever an index appears as an upper index as well as a lower index in the same term, then it is automatically summed over. For instance, in the expression \(\Gamma^l_{ij}g_{kl}\) the index \(l\) occurs both as an upper index and a lower index, hence we have \[\Gamma^l_{ij}g_{kl}=\Gamma^1_{ij}g_{k1}+\Gamma^2_{ij}g_{k2}.\]
Using the Einstein summation convention, the above equations can be written as \[\partial_{k}F\cdot \partial^2_{ij}F=\Gamma^l_{ij}g_{kl}.\] Now notice that for \(1\leqslant i,j,k\leqslant 2\) we have \[\partial_ig_{jk}=\partial_i\left(\partial_j F\cdot \partial_k F\right)=\partial^2_{ij}F\cdot \partial_k F+\partial_j F \cdot \partial^2_{ik}F.\] From this we compute \[\begin{gathered} \partial_{i}g_{jk}+\partial_jg_{ik}-\partial_k g_{ij}=\partial^2_{ij}F\cdot \partial_k F+\partial_j F \cdot \partial^2_{ik}F+\partial^2_{ji}F\cdot \partial_k F+\partial_i F \cdot \partial^2_{jk}F\\ -\partial^2_{ki} F \cdot \partial_j F-\partial_iF\cdot \partial^2_{kj}F \end{gathered}\] so that \[\partial_{i}g_{jk}+\partial_jg_{ik}-\partial_k g_{ij}=2\partial_k F \cdot \partial^2_{ij}F,\] where we use that \(\partial^2_{ij}F=\partial^2_{ji}F.\) In summary we have \[\tag{4.2} \Gamma^l_{ij}g_{kl}=\frac{1}{2}\left(\partial_{i}g_{jk}+\partial_jg_{ik}-\partial_k g_{ij}\right).\] Recall that we write \(g^{-1} : U \to M_{2,2}(\mathbb{R})\) for the map which assigns to \(q \in U\) the inverse of the matrix \(g(q).\) It is customary to write \[g^{-1}=\begin{pmatrix} g^{11} & g^{12} \\ g^{21} & g^{22}\end{pmatrix}\] for functions \(g^{ij} : U \to \mathbb{R}\) which satisfy \(g^{12}=g^{21}.\) By definition, we have \[g^{rk}g_{kl}=g^{r1}g_{1l}+g^{r2}g_{2l}=\left\{\begin{array}{cc} 1, & r=l, \\ 0, & r \neq l.\end{array}\right.\] Using this we can compute \[g^{rk}\Gamma^l_{ij}g_{kl}=\Gamma^l_{ij}g^{rk}g_{kl}=\Gamma^r_{ij}.\] Finally, using (4.2) we thus have \[\Gamma^r_{ij}=\frac{1}{2}g^{rk}\left(\partial_{i}g_{jk}+\partial_jg_{ik}-\partial_k g_{ij}\right).\]
Definition 4.2 • Christoffel symbols
The functions \(\Gamma^l_{ij} : U \to \mathbb{R}\) defined by \[\Gamma^{l}_{ij}=\frac{1}{2}g^{lk}\left(\partial_i g_{jk}+\partial_j g_{ik}-\partial_k g_{ij}\right)\qquad 1\leqslant i,j,k,l\leqslant 2.\] are the Christoffel symbols associated to the immersion \(F : U \to \mathbb{R}^3.\)
Notice that the Christoffel symbols satisfy \[\Gamma^l_{ij}=\Gamma^l_{ji}.\]
Example 4.3(Example 3.71
Example 3.71 • Surface of revolution ➔
continued). For a surface of revolution, we computed that \[g(q)=\begin{pmatrix} \gamma_1(v)^2 & 0 \\ 0 & 1 \end{pmatrix}\] so that \[g^{-1}(q)=\begin{pmatrix} \frac{1}{\gamma_1(v)^2} & 0 \\ 0 & 1 \end{pmatrix}.\] Moreover, we have \[\partial_2 g_{11}(q)=2\gamma_1(v)\gamma_1^{\prime}(v)\] and \(\partial_ig_{jk}=0\) otherwise. It follows that \[\Gamma^1_{11}=\Gamma^1_{22}=\Gamma^2_{12}=\Gamma^2_{21}=\Gamma^2_{22}=0\] and \[\begin{aligned}
\Gamma^1_{12}(q)&=\Gamma^1_{21}(q)=\frac{1}{2}g^{11}(q)\left(\partial_1g_{12}(q)+\partial_2g_{11}(q)-\partial_1 g_{12}(q)\right)\\
&=\frac{1}{2}g^{11}(q)\partial_2 g_{11}(q)=\frac{2\gamma_1(v)\gamma_1^{\prime}(v)}{2\gamma_1(v)^2}=\frac{\gamma^{\prime}_1(v)}{\gamma_1(v)}
\end{aligned}\] as well as \[\begin{aligned}
\Gamma^2_{11}(q)&=\frac{1}{2}g^{22}(q)\left(\partial_{1}g_{12}(q)+\partial_1g_{12}(q)-\partial_2g_{11}(q)\right)\\
&=-\frac{1}{2}g^{22}(q)\partial_2g_{11}(q)=-\frac{1}{2}2\gamma_1(v)\gamma_1^{\prime}(v)=-\gamma_1(v)\gamma_1^{\prime}(v).
\end{aligned}\]Let \(I\) be an interval and \(\gamma=(\gamma_1,\gamma_2) : I \to \mathbb{R}^2\) a smooth injective immersed curve with \(\gamma_1(t)>0\) for all \(t \in I.\) Consider \(F : [0,2\pi) \times I \to \mathbb{R}^3\) defined by \[(u,v)\mapsto f_{\mathbf{R}_{u}}(\gamma_1(v),0,\gamma_2(v))=(\gamma_1(v)\cos(u),\gamma_1(v)\sin(u),\gamma_2(v))\] where \(\mathbf{R}_u\) is the matrix corresponding to rotation around the \(z\)-axis with angle \(u\) \[\mathbf{R}_u=\begin{pmatrix} \cos(u) & -\sin(u) & 0 \\ \sin(u) & \cos(u) & 0 \\ 0 & 0 & 1\end{pmatrix}.\] Then, one can easily check that \(M=\operatorname{Im}(F)\subset \mathbb{R}^3\) is an immersed surface known as a surface of revolution. We compute the Gauss and mean curvature in the case where \(\gamma\) is a unit speed curve. We have \[\partial_1F(q)=\begin{pmatrix} -\gamma_1(v)\sin(u) \\ \gamma_1(v)\cos(u) \\ 0 \end{pmatrix} \qquad \text{and} \qquad \partial_2F(q)=\begin{pmatrix} \gamma^{\prime}_1(v)\cos(u) \\ \gamma_1^{\prime}(v)\sin(u) \\ \gamma_2^{\prime}(v)\end{pmatrix}.\] from which we compute \[g(q)=\begin{pmatrix} \gamma_1(v)^2 & 0 \\ 0 & 1 \end{pmatrix} \qquad \text{and} \qquad G(q)=\begin{pmatrix}\cos(u)\gamma_2^{\prime}(v) \\ \sin(u)\gamma_2^{\prime}(v) \\ -\gamma_1^{\prime}(v)\end{pmatrix}\] as well as \[A(q)=\begin{pmatrix} -\gamma_1(v)\gamma_2^{\prime}(v) & 0 \\ 0 & \gamma_1^{\prime\prime}(v)\gamma_2^{\prime}(v)-\gamma_1^{\prime}(v)\gamma_2^{\prime\prime}(v)\end{pmatrix}.\] Hence we obtain \[K(q)=\frac{\gamma_2^{\prime}(v)\left(\gamma_1^{\prime}(v)\gamma_2^{\prime\prime}(v)-\gamma_1^{\prime\prime}(v)\gamma_2^{\prime}(v)\right)}{\gamma_1(v)}\] Differentiating \[\gamma_1^{\prime}(v)^2+\gamma_2^{\prime}(v)^2=1\] with respect to \(v\) we deduce \[\gamma_1^{\prime}(v)\gamma_1^{\prime\prime}(v)=-\gamma_2^{\prime}(v)\gamma_2^{\prime\prime}(v),\] so that \[K(q)=\frac{-\gamma_1^{\prime\prime}(v)\gamma_1^{\prime}(v)^2-\gamma_1^{\prime\prime}(v)\gamma_2^{\prime}(v)^2}{\gamma_1(v)}=-\frac{\gamma_1^{\prime\prime}(v)}{\gamma_1(v)}.\] For the mean curvature we obtain \[H(q)=\frac{1}{2}\left(\frac{\gamma_2^{\prime}(v)}{\gamma_1(v)}+\gamma_1^{\prime}(v)\gamma^{\prime\prime}_2(v)-\gamma^{\prime}_2(v)\gamma_1^{\prime\prime}(v)\right).\] Notice that if \(\kappa : I \to \mathbb{R}\) denotes the signed curvature of the plane curve \(\gamma=(\gamma_1,\gamma_2) : I \to \mathbb{R}^2,\) then we can write \[H(q)=\frac{1}{2}\left(\kappa(v)+\frac{\gamma_2^{\prime}(v)}{\gamma_1(v)}\right).\]
Example 4.4(Example 3.68
Example 3.68 • Helicoid ➔
continued). For the Helicoid we compute that \[g(q)=\begin{pmatrix} 1 & 0 \\ 0 & 1+u^2\end{pmatrix}\] so that \[g^{-1}(q)=\begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{1+u^2} \end{pmatrix}.\] Moreover, we have \[\partial_1 g_{22}(q)=2u\] and \(\partial_i g_{jk}\) otherwise. It follows that \[\Gamma^2_{22}=\Gamma^2_{11}=\Gamma^1_{21}=\Gamma^1_{12}=\Gamma^1_{11}=0\] and \[\begin{aligned}
\Gamma^2_{12}(q)&=\Gamma^2_{21}(q)=\frac{1}{2}g^{22}(q)\left(\partial_1 g_{22}(q)+\partial_2 g_{12}(q)-\partial_2 g_{12}(q))\right)\\
&=\frac{1}{2}g^{22}(q)\partial_1 g_{22}(q)=\frac{2u}{2(1+u^2)}=\frac{u}{1+u^2},
\end{aligned}\] as well as \[\begin{aligned}
\Gamma^1_{22}&=\frac{1}{2}g^{11}(q)\left(2\partial_2 g_{12}(q)-\partial_1 g_{22}(q)\right)=-\frac{1}{2}g^{11}(q)\partial_1g_{22}(q)\\
&=-\frac{1}{2}2u=-u.
\end{aligned}\]Consider \[F : \mathbb{R}^2 \to \mathbb{R}^3, \qquad q=(u,v) \mapsto F(q)=(u\cos(v),u\sin(v),v)\] Clearly, \(F\) is smooth and injective and a calculation shows that \(F\) is an immersion, hence \(M=F(\mathbb{R}^2)\subset \mathbb{R}^3\) is an immersed surface called the Helicoid. Here we compute \[g(q)=\begin{pmatrix} 1 & 0 \\ 0 & 1+u^2\end{pmatrix} \qquad \text{and} \qquad G(q)=\frac{1}{\sqrt{1+u^2}}\begin{pmatrix} \sin(v) \\ -\cos(v) \\ u \end{pmatrix}\] as well as \[A(q)=\begin{pmatrix} 0 & -\frac{1}{\sqrt{1+u^2}} \\ -\frac{1}{\sqrt{1+u^2}} & 0 \end{pmatrix}.\] Which gives \[K(q)=-\frac{1}{(1+u^2)^2} \qquad \text{and} \qquad H(q)=0.\]
We now return to our problem of determining integrability conditions for finding \(F : U \to \mathbb{R}^3\) when we are given the functions \(g_{ij},A_{ij}\) on \(U.\) Using the summation convention, we can write (4.1) as \[\partial^2_{jk}F=A_{jk}G+\Gamma^l_{jk}\partial_lF.\] Using this we compute \[\begin{aligned} Q_{jikm}:&=\left[\partial_i\left(\partial^2_{jk}F\right)\right]\cdot \partial_m F=\left[\partial_i\left(A_{jk}G+\Gamma^l_{jk}\partial_l F\right)\right]\cdot \partial_m F\\ &=\partial_iA_{jk}\underbrace{G\cdot \partial_m F}_{=0}+A_{jk}\underbrace{\partial_iG\cdot \partial_m F}_{=-A_{im}}+\partial_i\Gamma^l_{jk}\underbrace{\partial_lF\cdot \partial_m F}_{=g_{lm}}+\Gamma^l_{jk}\underbrace{\partial^2_{il}F\cdot \partial_m F}_{=\Gamma^r_{il}g_{mr}}\\ &=-A_{jk}A_{im}+\partial_i\Gamma^l_{jk}g_{lm}+\Gamma^l_{jk}\Gamma^r_{il}g_{mr}. \end{aligned}\] Indices that are summed over can be given new “names”, so that \[\Gamma^l_{jk}\Gamma^r_{il}g_{mr}=\Gamma^a_{jk}\Gamma^r_{ia}g_{mr}=\Gamma^a_{jk}\Gamma^b_{ia}g_{mb}=\Gamma^r_{jk}\Gamma^b_{ir}g_{mb}=\Gamma^r_{jk}\Gamma^l_{ir}g_{ml}.\] Since \(g_{ml}=g_{lm}\) we thus obtain \[Q_{jikm}=-A_{jk}A_{im}+\left(\partial_i\Gamma^l_{jk}+\Gamma^r_{jk}\Gamma^l_{ir}\right)g_{lm}.\] Using that third derivatives commute, we have \(\partial_j(\partial^2_{ik}F)=\partial_i(\partial^2_{jk}F)\) and hence \[0=Q_{ijkm}-Q_{jikm}=\left(\partial_j\Gamma^l_{ik}-\partial_i\Gamma^l_{jk}+\Gamma^r_{ik}\Gamma^l_{jr}-\Gamma^r_{jk}\Gamma^l_{ir}\right)g_{lm}-(A_{ik}A_{jm}-A_{jk}A_{im}).\] Writing \[\tag{4.3} R_{ijkm}=\left(\partial_j\Gamma^l_{ik}-\partial_i\Gamma^l_{jk}+\Gamma^r_{ik}\Gamma^l_{jr}-\Gamma^r_{jk}\Gamma^l_{ir}\right)g_{lm},\] we have the so-called Gauss equations \[\tag{4.4} R_{ijkm}=A_{ik}A_{jm}-A_{jk}A_{im},\] which must hold for all \(1\leqslant i,j,k,m\leqslant 2.\) The functions \(R_{ijkm}\) depend on the \(g_{lm}\) and the Christoffel symbols only, thus they can be computed from knowing the functions \(g_{ij}.\) If we are given functions \(g_{ij},A_{ij}\) on \(U,\) then the Gauss equations are necessary conditions for the existence of an immersion \(F : U \to \mathbb{R}^3\) realising \(g_{ij},A_{ij}.\) We can derive more necessary conditions as follows: Consider \[\tag{4.5} \begin{aligned} P_{jik}:&=\left[\partial_i\left(\partial^2_{jk} F\right)\right]\cdot G=\left[\partial_i\left(A_{jk}G+\Gamma^l_{jk}\partial_l F\right)\right]\cdot G\\ &=\partial_{i}A_{jk}\underbrace{G\cdot G}_{=1}+A_{jk}\partial_i G\cdot G+\partial_i\Gamma^l_{jk}\underbrace{\partial_l F\cdot G}_{=0}+\Gamma^l_{jk}\partial^2_{il}F\cdot G. \end{aligned}\] Since \(G(q)\cdot G(q)=1,\) it follows as before that \(\partial_iG(q)\cdot G(q)=0\) for all \(q \in U\) and \(i=1,2.\) Therefore (4.5) together with (4.1) gives \[P_{jik}=\partial_iA_{jk}+\Gamma^l_{jk}\left(\Gamma^m_{il}\partial_m F+A_{il}G\right)\cdot G=\partial_iA_{jk}+\Gamma^l_{jk}A_{il}.\] Again, using that third partial derivatives commute, we arrive at \[0=P_{ijk}-P_{jik}=\partial_jA_{ik}-\partial_iA_{jk}+\Gamma^l_{ik}A_{jl}-\Gamma^l_{jk}A_{il}.\] Equivalently, at the so-called Codazzi equations \[\tag{4.6} \partial_jA_{ik}-\partial_iA_{jk}=\Gamma^l_{jk}A_{il}-\Gamma^l_{ik}A_{jl},\] which must hold for all \(1\leqslant i,j,k\leqslant 2.\) This shows:
Proposition 4.5 • Gauss–Codazzi equations
Let \(U\subset \mathbb{R}^2\) be an open subset and \(g_{ij},A_{ij} : U \to \mathbb{R}\) smooth functions for \(1\leqslant i,j\leqslant 2\) with \(g_{12}=g_{21}\) and \(A_{12}=A_{21}.\) Then the Gauss – and Codazzi equations \[\tag{4.7} R_{ijkm}=A_{ik}A_{jm}-A_{jk}A_{im} \qquad \text{and} \qquad \partial_jA_{ik}-\partial_iA_{jk}=\Gamma^l_{jk}A_{il}-\Gamma^l_{ik}A_{jl},\] are necessary conditions for the existence of a smooth immersion \(F : U \to \mathbb{R}^3\) whose associated functions via (3.9) and (3.12) are \(g_{ij}\) and \(A_{ij}.\)
Remark 4.6
A theorem which goes beyond the scope of this course states that if \(U\) is so-called simply connected (which is in particular the case if \(U\) is a rectangle), then the equations (4.7) are also sufficient, provided \(g_{ij}(q)\) is positive definite for all \(q \in U.\) That is, if \(g_{ij},A_{ij}\) are given functions on \(U\) satisfying (4.7) and \(g_{ij}\) is positive definite, then there exists an immersion \(F : U \to \mathbb{R}^3\) realising \(g_{ij}\) and \(A_{ij}\) and moreover, the image of this immersion if unique up to post composition by a Euclidean motion.
Loosely speaking this all states that the functions \(g_{ij}\) and \(A_{ij}\) capture an immersed surface up to Euclidean motion.
In the case of a curve in \(\mathbb{R}^2,\) we saw that the signed curvature captures the curve up to Euclidean motion. We can prescribe any smooth function as the signed curvature of a plane curve, whereas in the case of a surface the functions \(g_{ij},A_{ij}\) that we prescribe must satisfy the integrability conditions (4.7).
4.2 The covariant derivative revisited
Recall that \[\begin{pmatrix} g_{11}(q) & g_{12}(q) \\ g_{12}(q) & g_{22}(q)\end{pmatrix} \qquad \text{and} \qquad \begin{pmatrix} A_{11}(q) & A_{12}(q) \\ A_{12}(q) & A_{22}(q)\end{pmatrix}\] are the matrix representations of the first – and second – fundamental form \(\mathrm{I}_{F(q)}, \mathrm{I}\!\,\mathrm{I}_{F(q)}\) at \(F(q)\) with respect to the ordered basis \(((\partial_1 F(q))_{F(q)},(\partial_2 F(q))_{F(q)})\) of \(T_{F(q)}M,\) respectively. It is natural to wonder whether the Christoffel symbols \(\Gamma^i_{jk}\) also encode a natural map.
Recall that if \(M\subset \mathbb{R}^3\) is an embedded surface and \(\gamma : I \to M\) a smooth curve and \(X : I \to TM\) a vector field along \(\gamma,\) so that \(X(t) \in T_{\gamma(t)}M,\) then we defined the covariant derivative of \(X\) as \[\frac{\mathrm{D}X}{\mathrm{d}t}(t)=\Pi^{\perp}_{T_{\gamma(t)}M}(\dot{X}(t))\] for all \(t \in I.\) Phrased differently, \(\frac{\mathrm{D}X}{\mathrm{d}t}(t)\) is the tangential component of the vector \(\dot{X}(t) \in T_{\gamma(t)}M\) with respect to the direct sum decomposition \[T_{\gamma(t)}\mathbb{R}^3=T_{\gamma(t)}M\oplus T_{\gamma(t)}M^{\perp}.\] We can use the covariant derivative of a vector field along a curve to define a directional derivative of a vector field. If \(Y : M \to TM\) is a smooth vector field on \(M\) and \(\vec{v}_p \in TM,\) we define the derivative of the vector field \(Y\) in the tangent direction \(\vec{v}_p\) by \[\nabla_{\vec{v}_p}Y:=\frac{\mathrm{D}}{\mathrm{d}t}(Y\circ \gamma)(0),\] where \(\gamma : (-\epsilon,\epsilon) \to M\) is a smooth curve with \(\gamma(0)=p,\) \(\dot{\gamma}(0)=\vec{v}_p\) and \(\epsilon>0.\) We have to make sure that the choice of \(\gamma\) does not matter, that is \(\nabla_{\vec{v}_p}Y\) does only depend on \(Y\) and \(\vec{v}_p.\) Write \[Y(p)=\begin{pmatrix}Y^1(p) \\ Y^2(p) \\ Y^3(p) \end{pmatrix}_p\] for smooth functions \(Y^i : M \to \mathbb{R}.\) Then by definition \[\nabla_{\vec{v}_p}Y=\Pi^{\perp}_{T_pM}(\dot{Z}(0)),\] where \(Z=Y\circ \gamma.\) We have \[\dot{Z}(0)=\begin{pmatrix} (Y^1\circ \gamma)^{\prime}(0) \\ (Y^2\circ \gamma)^{\prime}(0) \\ (Y^3\circ \gamma)^{\prime}(0) \end{pmatrix}_{\gamma(0)}=\begin{pmatrix} \mathrm{d}Y^1(\dot{\gamma}(0)) \\ \mathrm{d}Y^2(\dot{\gamma}(0)) \\ \mathrm{d}Y^3(\dot{\gamma}(0)) \end{pmatrix}_{\gamma(0)}=\begin{pmatrix} \mathrm{d}Y^1(\vec{v}_p) \\ \mathrm{d}Y^2(\vec{v}_p) \\ \mathrm{d}Y^3(\vec{v}_p) \end{pmatrix}_{p}.\] We conclude that \(\dot{Z}(0)\) does only depend on \(Y\) and \(\vec{v}_p,\) and hence so does \(\nabla_{\vec{v}_p}Y,\) since \(\Pi^\perp_{T_pM}\) does not depend on the choice of curve \(\gamma.\)
Let \(\mathfrak{X}(M)\) denote the set of smooth vector fields on \(M.\) We define addition in the natural way, that is, for \(X,Y \in \mathfrak{X}(M),\) we define for all \(p \in M\) \[(X+Y)(p):=X(p)+Y(p).\] Moreover, for a smooth function \(f : M \to \mathbb{R}\) we define \[(fX)(p):=f(p)X(p).\] For two vector fields \(X,Y \in \mathfrak{X}(M)\) and all \(p \in M,\) we define \[(\nabla_XY)(p):=\nabla_{X(p)}Y \in T_pM.\] With these rules in place we can think of \(\nabla\) as a map \(\nabla : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M)\) defined by \[(X,Y) \mapsto \nabla_XY\] The map \(\nabla\) is also called the covariant derivative.
Lemma 4.7
Let \(X_1,X_2,Y_1,Y_2 \in \mathfrak{X}(M)\) and \(f : M \to \mathbb{R}\) a smooth function. Then the covariant derivative satisfies:
\(\nabla_{X_1+X_2} Y_1=\nabla_{X_1}Y_1+\nabla_{X_2}Y_1\);
\(\nabla_{X_1}(Y_1+Y_2)=\nabla_{X_1}Y_1+\nabla_{X_2}Y_2\);
\(\nabla_{fX_1}Y_1=f\nabla_{X_1}Y_1\);
\(\nabla_{X_1}(fY_1)=f\nabla_{X_1}Y_{1}+\mathrm{d}f(X_1)Y_1.\)
Proof. Exercise.
Remark 4.8 In Lemma 4.7
Lemma 4.7 ➔
, \(\mathrm{d}f(X_1)\) is the smooth function on \(M\) defined by \[\mathrm{d}f(X_1) : M \to \mathbb{R}, \qquad p \mapsto \mathrm{d}f(X_1(p)).\]Let \(X_1,X_2,Y_1,Y_2 \in \mathfrak{X}(M)\) and \(f : M \to \mathbb{R}\) a smooth function. Then the covariant derivative satisfies:
\(\nabla_{X_1+X_2} Y_1=\nabla_{X_1}Y_1+\nabla_{X_2}Y_1\);
\(\nabla_{X_1}(Y_1+Y_2)=\nabla_{X_1}Y_1+\nabla_{X_2}Y_2\);
\(\nabla_{fX_1}Y_1=f\nabla_{X_1}Y_1\);
\(\nabla_{X_1}(fY_1)=f\nabla_{X_1}Y_{1}+\mathrm{d}f(X_1)Y_1.\)
By construction the covariant derivative \(\nabla\) does depend on the first fundamental form only, it is thus an object of intrinsic surface geometry. In fact, the Christoffel symbols do encode \(\nabla,\) more precisely, we have the following statement:
Proposition 4.9
Let \(M\subset \mathbb{R}^3\) be a surface and \(F : U \to M\) a local parametrisation of \(M\) with Christoffel symbols \(\Gamma^k_{ij} : U \to \mathbb{R}\) for \(i,j,k=1,2.\) Then on \(\operatorname{Im}(F)\subset M\) we obtain vector fields \(B_i\) for \(i=1,2\) defined by the rule \[B_i(F(q))=\left(\partial_i F(q)\right)_{F(q)}\] for all \(q \in U.\) For these vector fields we have \[(\nabla_{B_i}B_j)(F(q))=\Gamma^k_{ij}(q)B_k(F(q))\] for all \(q \in U\) and where we employ the summation convention.
Remark 4.10
Since \(F : U \to M\) is an immersion, it follows that \(\{B_1(p),B_2(p)\}\) is a basis of \(T_pM\) for all \(p \in F(U).\)
For the proof we need the following:
Lemma 4.11
Let \(M\subset \mathbb{R}^3\) be a surface and \(F : U \to M\) a local parametrisation of \(M.\) Suppose \(c=(c^1,c^2) : I \to U\) is a smooth curve and \(X : I \to M\) is a vector field along the curve \(\gamma=F\circ c : I \to M.\) Writing \[\tag{4.8} X(t)=X^j(t)B_j(\gamma(t))\] for unique smooth functions \(X^i : I \to \mathbb{R},\) we have \[\tag{4.9} \frac{\mathrm{D}X}{\mathrm{d}t}(t)=\left(\frac{\mathrm{d}X^l}{\mathrm{d}t}(t)+\Gamma^l_{ij}(c(t))X^j(t)\frac{\mathrm{d}c^i}{\mathrm{d}t}(t)\right)B_l(\gamma(t)),\] where in (4.8) and (4.9) we employ the summation convention.
Proof. Taking the time derivative of (4.8), we obtain \[\dot{X}(t)=\left(\frac{\mathrm{d}X^j}{\mathrm{d}t}(t)\partial_jF(c(t))+X^j(t)\frac{\mathrm{d}c^j}{\mathrm{d}t}(t)\partial^2_{ij}F(c(t))\right)_{\gamma(t)}.\] Since \(\partial^2_{ji}F=\Gamma^l_{ji}\partial_lF+A_{ji}G,\) we get \[\begin{gathered} \dot{X}(t)=\left(\frac{\mathrm{d}X^j}{\mathrm{d}t}(t)\partial_jF(c(t))\right.\\\left.+X^j(t)\frac{\mathrm{d}c^i}{\mathrm{d}t}(t)\left(\Gamma^l_{ij}(c(t))\partial_lF(c(t))+A_{ij}(c(t))G(c(t))\right)\right)_{\gamma(t)} \end{gathered}\] The tangential component of \(\dot{X}(t)\) is thus given by \[\frac{\mathrm{D}X}{\mathrm{d}t}(t)=\left(\frac{\mathrm{d}X^l}{\mathrm{d}t}(t)+\Gamma^l_{ij}(c(t))X^j(t)\frac{\mathrm{d}c^i}{\mathrm{d}t}(t)\right)B_l(\gamma(t)),\] as claimed.
Proposition 4.9 ➔
. In order to compute \((\nabla_{B_i}B_j)(F(q))\) we can choose a curve \(\gamma : (-\epsilon,\epsilon) \to M\) so that \(\dot{\gamma}(0)=B_i(F(q))\) and then evaluate \(\frac{\mathrm{D}}{\mathrm{d}t}X(0)\) using the formula (4.9) for \(X:=B_j\circ \gamma.\) Let \(e_1,e_2\) denote the standard basis of \(\mathbb{R}^2,\) interpreted as points, and consider the curve \(c : (-\epsilon,\epsilon) \to U\) defined by the rule \(c(t)=q+te_i\) for \(\epsilon\) sufficiently small. Notice that \(\gamma=F\circ c\) satisfies \(\gamma(0)=F(q)\) and moreover \(\dot{\gamma}(0)=B_i(F(q)).\) Since \[X(t)=B_j(\gamma(t))=\Big(\partial_jF(c(t))\Big)_{\gamma(t)}\] it follows that the functions \(X^i\) in (4.8) are given by \(X^i(t)=\delta^i_j\) for all \(t \in (-\epsilon,\epsilon).\) Moreover, we also have \[\frac{\mathrm{d}c^k}{\mathrm{d}t}(t)=\delta^k_i.\] Renaming indices in (4.9) and evaluating at \(t=0\) we obtain \[\begin{aligned}
(\nabla_{B_i}B_j)(F(q))&=\frac{\mathrm{D}X}{\mathrm{d}t}(0)=\left(\frac{\mathrm{d}X^l}{\mathrm{d}t}(0)+\Gamma^l_{rm}(c(0))X^m(0)\frac{\mathrm{d}c^r}{\mathrm{d}t}(0)\right)\Big(\partial_lF(c(0))\Big)_{\gamma(0)}\\
&=\left(\Gamma^l_{rm}(q)\delta^m_j\delta^r_i\right)\Big(\partial_lF(q)\Big)_{F(q)}=\Gamma^l_{ij}(q)B_l(F(q)),
\end{aligned}\] as claimed.Let \(M\subset \mathbb{R}^3\) be a surface and \(F : U \to M\) a local parametrisation of \(M\) with Christoffel symbols \(\Gamma^k_{ij} : U \to \mathbb{R}\) for \(i,j,k=1,2.\) Then on \(\operatorname{Im}(F)\subset M\) we obtain vector fields \(B_i\) for \(i=1,2\) defined by the rule \[B_i(F(q))=\left(\partial_i F(q)\right)_{F(q)}\] for all \(q \in U.\) For these vector fields we have \[(\nabla_{B_i}B_j)(F(q))=\Gamma^k_{ij}(q)B_k(F(q))\] for all \(q \in U\) and where we employ the summation convention.
Proposition 4.9 ➔
we thus obtain:Let \(M\subset \mathbb{R}^3\) be a surface and \(F : U \to M\) a local parametrisation of \(M\) with Christoffel symbols \(\Gamma^k_{ij} : U \to \mathbb{R}\) for \(i,j,k=1,2.\) Then on \(\operatorname{Im}(F)\subset M\) we obtain vector fields \(B_i\) for \(i=1,2\) defined by the rule \[B_i(F(q))=\left(\partial_i F(q)\right)_{F(q)}\] for all \(q \in U.\) For these vector fields we have \[(\nabla_{B_i}B_j)(F(q))=\Gamma^k_{ij}(q)B_k(F(q))\] for all \(q \in U\) and where we employ the summation convention.
Corollary 4.12
Let \(M\subset \mathbb{R}^3\) be a surface and \(F : U \to M\) a local parametrisation of \(M.\) Suppose \(c=(c^1,c^2) : I \to U\) is a smooth curve. Then \(\gamma=F\circ c : I \to M\) is a geodesic if and only if \(c\) satisfies the so-called geodesic equation \[\tag{4.10} \frac{\mathrm{d}^2 c^l}{\mathrm{d}t^2}(t)+\Gamma^l_{ij}(c(t))\frac{\mathrm{d}c^i}{\mathrm{d}t}(t)\frac{\mathrm{d}c^j}{\mathrm{d}t}(t)=0\] for all \(t \in I\) and where we employ the summation convention.
Proof. This follows immediately from (4.9) for \(X=\dot{\gamma}\) so that \(X^i(t)=\frac{\mathrm{d}c^i}{\mathrm{d}t}(t).\)
Also, for vector fields \(X,Y \in \mathfrak{X}(M)\) we obtain a function \[\langle X,Y\rangle : M \to \mathbb{R}, \qquad p \mapsto \langle X(p),Y(p)\rangle_p\] and with this definition we have for all \(Z \in \mathfrak{X}(M)\) \[\tag{4.11} \mathrm{d}\left(\langle X,Y\rangle\right)(Z)=\langle \nabla_ZX,Y\rangle+\langle X,\nabla_ZY\rangle,\] which can again be deduced from (4.9).
Example 4.13(Geodesics on the helicoid – Example 4.4
Example 4.4 ➔
continued). For explicit calculations it is often convenient to write \(u(t)\) instead of \(c^1(t)\) and \(v(t)\) instead of \(c^2(t).\) Doing so we obtain for the geodesic equation on the helicoid \[\begin{aligned}
0&=\frac{\mathrm{d}^2 u}{\mathrm{d}t^2}(t)-u(t)\frac{\mathrm{d}v}{\mathrm{d}t}(t)\frac{\mathrm{d}v}{\mathrm{d}t}(t),\\
0&=\frac{\mathrm{d}^2 v}{\mathrm{d}t^2}(t)+\frac{2u(t)}{1+u(t)^2}\frac{\mathrm{d}u}{\mathrm{d}t}(t)\frac{\mathrm{d}v}{\mathrm{d}t}(t)
\end{aligned}\] or, using primes to indicate derivatives and omitting writing \(t,\) we have \[u^{\prime\prime}=uv^{\prime}v^{\prime} \qquad\text{and} \qquad v^{\prime\prime}=-\frac{2u}{1+u^2}u^{\prime}v^{\prime}.\](Example 3.68 continued). For the Helicoid we compute that \[g(q)=\begin{pmatrix} 1 & 0 \\ 0 & 1+u^2\end{pmatrix}\] so that \[g^{-1}(q)=\begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{1+u^2} \end{pmatrix}.\] Moreover, we have \[\partial_1 g_{22}(q)=2u\] and \(\partial_i g_{jk}\) otherwise. It follows that \[\Gamma^2_{22}=\Gamma^2_{11}=\Gamma^1_{21}=\Gamma^1_{12}=\Gamma^1_{11}=0\] and \[\begin{aligned} \Gamma^2_{12}(q)&=\Gamma^2_{21}(q)=\frac{1}{2}g^{22}(q)\left(\partial_1 g_{22}(q)+\partial_2 g_{12}(q)-\partial_2 g_{12}(q))\right)\\ &=\frac{1}{2}g^{22}(q)\partial_1 g_{22}(q)=\frac{2u}{2(1+u^2)}=\frac{u}{1+u^2}, \end{aligned}\] as well as \[\begin{aligned} \Gamma^1_{22}&=\frac{1}{2}g^{11}(q)\left(2\partial_2 g_{12}(q)-\partial_1 g_{22}(q)\right)=-\frac{1}{2}g^{11}(q)\partial_1g_{22}(q)\\ &=-\frac{1}{2}2u=-u. \end{aligned}\]
Example 4.14 • Geodesics on the torus We can parametrise the torus in terms of \[F : (0,2\pi)\times (0,2\pi r), \qquad (u,v) \mapsto (\cos(u)\gamma_1(v),\sin(u)\gamma_1(v),\gamma_2(v))\] where \(\gamma_1(v)=R+r\cos(v/r)\) and \(\gamma_2(v)=r\sin(v/r).\) Since \(\gamma=(\gamma_1,\gamma_2) : (0,2\pi r) \to \mathbb{R}\) is a unit speed curve, we obtain for the geodesic equation \[u^{\prime\prime}=-2\frac{\gamma_1^{\prime}(v)}{\gamma_1(v)}u^{\prime}v^{\prime}=2\frac{\sin(v/r)}{R+r\cos(v/r)}u^{\prime}v^{\prime}\] and \[v^{\prime\prime}=\gamma_1(v)\gamma_1^{\prime}(v)u^{\prime}u^{\prime}=-(R+r\cos(v/r))\sin(v/r)u^{\prime}u^{\prime},\] where we use the identities from Example 4.3
Example 4.3 ➔
.(Example 3.71 continued). For a surface of revolution, we computed that \[g(q)=\begin{pmatrix} \gamma_1(v)^2 & 0 \\ 0 & 1 \end{pmatrix}\] so that \[g^{-1}(q)=\begin{pmatrix} \frac{1}{\gamma_1(v)^2} & 0 \\ 0 & 1 \end{pmatrix}.\] Moreover, we have \[\partial_2 g_{11}(q)=2\gamma_1(v)\gamma_1^{\prime}(v)\] and \(\partial_ig_{jk}=0\) otherwise. It follows that \[\Gamma^1_{11}=\Gamma^1_{22}=\Gamma^2_{12}=\Gamma^2_{21}=\Gamma^2_{22}=0\] and \[\begin{aligned} \Gamma^1_{12}(q)&=\Gamma^1_{21}(q)=\frac{1}{2}g^{11}(q)\left(\partial_1g_{12}(q)+\partial_2g_{11}(q)-\partial_1 g_{12}(q)\right)\\ &=\frac{1}{2}g^{11}(q)\partial_2 g_{11}(q)=\frac{2\gamma_1(v)\gamma_1^{\prime}(v)}{2\gamma_1(v)^2}=\frac{\gamma^{\prime}_1(v)}{\gamma_1(v)} \end{aligned}\] as well as \[\begin{aligned} \Gamma^2_{11}(q)&=\frac{1}{2}g^{22}(q)\left(\partial_{1}g_{12}(q)+\partial_1g_{12}(q)-\partial_2g_{11}(q)\right)\\ &=-\frac{1}{2}g^{22}(q)\partial_2g_{11}(q)=-\frac{1}{2}2\gamma_1(v)\gamma_1^{\prime}(v)=-\gamma_1(v)\gamma_1^{\prime}(v). \end{aligned}\]
Remark 4.15
We refer to M12 for techniques to solve systems of ordinary differential equations. Generally speaking, it is rather exceptional that one can explicitly write down a solution to a geodesic equation.