Notes for Introduction to Differential Geometry
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Now we will study some global properties of Curves. Two classic results are: the isoperimetric inequality and the Four Vertex Problem.

In this course we will only focus on the former one. We will state the problem with curves and in the next section we will generalise the questions to higher dimensions.

The following result is known as the Inverse Function Theorem (IFT). It states the existence of a local diffeomorphism between two sets by requiring that the differential of a function between them is represented by a non-singular matrix.

The proof will rely on the following lemma:

Examples

• The graph of a paraboloid: \(S = \{ (x,y,z) : z = x^2 + y^2 \}\).
• The ellipsoid: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{b^2} = 1\).

• The sphere using the stereographic projection: \(X:V_1 \subset \mathbb{R}^2 \to \mathbb{R}^3\) given by

  \begin{equation} X(u,v) = \left( \frac{2u}{ u^2 + v^2 + 1}, \frac{2v}{u^2 + v^2 +1}, \frac{u^2 + v^2 -1}{ u^2 + v^2 +1} \right) \end{equation}

Take \(p\in \psi^{-1}(W)\) and put \(q = h(p)\). We write \(\varphi = (x,y,z)\) and \(\psi = (\overline{x},\overline{y},\overline{z})\).

I. Since \(\varphi\) is a parameterisation we can assume that

\begin{equation} \left| \begin{array}{cc} \frac{\partial x}{\partial u}(q) & \frac{\partial x}{\partial v}(q) \\ \frac{\partial y}{\partial u}(q) & \frac{\partial y}{\partial v}(q) \\ \end{array} \right| \neq 0. \end{equation}

II. Define an extension of \(\varphi\). Let \(F:U\times \mathbb{R} \to \mathbb{R}^3\)

\begin{equation} F(u,v,t,) = \varphi(x(u,v), y(u,v), z(u,v) + t) = \varphi(u,v) + t\vec{e}_3. \end{equation}

One can explicitly write \(dF_{q}\) and verify that \(\det(dF_{p}) \neq 0\), and the restriction of \(F\) to the set \(U\times \{0\}\) agrees with \(\varphi\):

\begin{equation} \left. F \right|_{U\times\{0\}} = \varphi. \end{equation}

III. Apply the IFT to \(F\) to conclude that there is a neighbourhood \(W_2 \subset \mathbb{R}^3\), \(\varphi(q) \in W_2\) and a neighbourhood \(U_2 \subset U\times \mathbb{R}\) such taht \(F: U_2 \to W_2\) is a diffeomorphism.

IV. Use the fact that \(\psi\) is an homomorphism, then there is a set \(V_2\subset V\) with \(p\in V_2\) and such that \(\psi(V_2) \subset W_2\).

Now, the restriction of \(h\) to \(V_2\) is such that

\begin{equation} \left. h \right|_{V_2} = \left. \left(F^{-1} \circ \psi\right) \right|_{V_2}. \end{equation}

Then \(h\) is differentiable at \(p\in \psi^{-1}(W)\) since it is the composition of two differentiable maps.

Remark. The previoius definition is independent of the parameterisation.

From our previous formulae, if \(S\) is a regular surface then the map from the surface to the unit sphere \(\nu : S \to \mathbb{S}^2\), is well defined. More over for any \(\vec{p} \in S\), the derivative is the linear map \(d\nu_p : T_pS \to T_{\nu(p)}\mathbb{S}^2\). It is not difficult to show that there is a natural identification

\begin{equation} T_{\nu(p)}\mathbb{S}^2 \cong T_pS, \end{equation}

hence we can re-write \(d\nu_p : T_{p}S \to T_{p}S\). Since \(\{\partial_1, \partial_2\}\) is a basis for \(T_pS\), we can wrtie in the coordinates \(\varphi = \varphi(x_1, x_2)\), the following system of equations

\begin{equation} \begin{split} \frac{\partial \nu}{\partial x_1} &= a_1^1 \partial_1 + a_1^2 \partial_2\\ \frac{\partial \nu}{\partial x_2} &= a_2^1 \partial_1 + a_2^2 \partial_2. \end{split} \end{equation}

In what follows we use the Einstein's notation and we simply wrtie

\begin{equation} \frac{\partial \nu}{\partial x_i} = a_{i}^{j}\partial_{j}. \end{equation}

Taking the inner product with \(\partial_k\) we write then

\begin{equation}\label{eq:weingart01} \left\langle \frac{\partial \nu}{\partial x_i}, \partial_k \right\rangle = a_{i}^{j}g_{jk}. \end{equation}

Note that, since \(\langle \nu , \partial_k \rangle = 0\), and writing \(\partial_k = \frac{\partial \varphi}{\partial x_k}\) we can show that

\begin{equation} \left\langle \frac{\partial \nu}{\partial x_i}, \partial_k \right\rangle = - \left\langle \nu, \frac{\partial^2 \varphi}{\partial x_i \partial x_k} \right\rangle = - A_{ik}. \end{equation}

The last equation together with \eqref{eq:weingart01} we have

\begin{equation}\label{eq:weingart02} -A_{ik} = a_{i}^{j}g_{jk}. \end{equation}

If we write the entries of the inverse matrix \((g^{-1})_{ij} := g^{ij}\), that is, using upper indices, then it follows from the equation \eqref{eq:weingart02} that

\begin{equation} a_i^j = - g^{jk}A_{ki}, \end{equation}

We have shown

The eigenvalues of the shape operator will give us more information about the curvature of the space.

Remark: If the curve is parameterise by arc-length, from the Frenet equations and the chain rule we have

\begin{equation} k_{n} = \langle \kappa N, \nu_{p}\rangle = \langle \frac{d}{dt}T, \nu_{p}\rangle = - \langle T, \frac{d}{dt}\nu_{p}\rangle = - \langle T, \frac{\partial}{\partial x_i}\nu_{p} \dot{x}_{i}(t)\rangle = \langle T, g^{jk}A_{ik}\partial_j \dot{x}_{i}(t)\rangle = - \left\langle T, S(T) \right\rangle, \end{equation}

where \(T = \dot{x}_i\partial_i\), and \(S(T) = \dot{x}_iT(\partial_i) = \dot{x}_i\frac{\partial v_{p}}{\partial_i}\).

Note the following. If \(\lambda_1 \geq \lambda_2\) are the eigenvalues of the shape operator with normalised and orthogonal eigenvectors \(\vec{v_1}, \vec{v}_2\), then for any curve with velocity vector \(\vec{w} = w_1\cos\theta \vec{v}_1 + w_2\sin\theta \vec{v}_2\) we have that

\begin{equation} \kappa_{n} = - \left\langle \vec{w}, S(\vec{w}) \right\rangle = - \left\langle \vec{w}, w_1\cos\theta S(\vec{v_1}) + w_2\cos\theta S(\vec{v_2}) \right\rangle = - \left\langle \vec{w}, \lambda_{1} w_1\cos\theta \vec{v_1} + \lambda_{2} w_2\cos\theta \vec{v}_2 \right\rangle = - \lambda_1 w_1^2\cos^2\theta - \lambda^2_2 w_2^2\sin^2\theta, \end{equation}

from this one can show that the maximum and minimum of the normal curvatures \(k_n\) are precisely the eigenvalues of the shape operator.

Look back at our definition of the second fundamental form 32.4. There, we were measuring the projection of the second derivatives of the parameterisation around \(\vec{p}\in S\), to the normal direction \(\nu\). What about the projection to the tangent space \(T_pS\)? Using again the Einstein's notation we can write that system of equations as

\begin{equation} \frac{\partial^2 \varphi}{\partial x_{i} \partial x_{j}} = \Gamma_{ij}^k \partial_{k} + A_{ij} \nu. \end{equation}

Note that

\begin{equation} \begin{split} \frac{\partial g_{ij}}{\partial x_l} & = \frac{\partial}{\partial x_l} \left\langle \frac{\partial\varphi}{\partial x_i}, \frac{\partial\varphi}{\partial x_j} \right\rangle \\ & =\left\langle \frac{\partial^2 \varphi}{\partial x_l \partial x_i}, \frac{\partial\varphi}{\partial x_j} \right\rangle + \left\langle \frac{\partial \varphi}{\partial x_i}, \frac{\partial^2\varphi}{\partial x_l \partial x_j} \right\rangle \\ & = \Gamma_{li}^{r}g_{rj} + \Gamma_{lj}^{r}g_{ri},\\ \frac{\partial g_{il}}{\partial x_j} & = \Gamma_{ji}^{r}g_{rl} + \Gamma_{jl}^{r}g_{ri},\\ \frac{\partial g_{jl}}{\partial x_i} & = \Gamma_{ij}^{r}g_{rl} + \Gamma_{il}^{r}g_{rj}. \end{split} \end{equation}

Since \(\Gamma_{ij}^k = \Gamma_{ji}^k\), then the proposition follows now easily.

We have seen that the expressions of the derivatives of \(\partial_1, \partial_2\) and \(\nu\) can be expressed in the basis \(\{\partial_1, \partial_2, \nu\}\), with coefficients involving the first fundamental form, its derivatives, and the second fundamental form.

If we want to try to get more relationships between these terms, we should take a look at equations involving third order derivatives of the local parameterisation like

\begin{equation} \frac{\partial}{\partial x_{k}} \left(\frac{\partial^2 \varphi}{\partial x_j \partial x_i}\right) - \frac{\partial}{\partial x_{j}} \left(\frac{\partial^2 \varphi}{\partial x_k \partial x_i}\right) = 0. \end{equation}

The proof is a straightforward application of the Gauss equation \eqref{eq:Gausseq}

\begin{equation} R_{1212} = A_{11} A_{22} - A_{12}^2 = \det(A). \end{equation}

Then

\begin{equation} K = \frac{\det(A)}{\det{g}} = \frac{R_{1212}}{\det(g)}, \end{equation}

which depends only on the first fundamental form and its derivatives.

To obtain a more explicit result we make use of

\begin{equation} R_{ijk}^l = \frac{\partial}{\partial x_j}(\Gamma_{ik}^l) - \frac{\partial}{\partial x_i}(\Gamma_{jk}^l) + \Gamma_{js}^l\Gamma_{ik}^s - \Gamma_{is}^l\Gamma_{jk}^s. \end{equation} \begin{equation} R_{1212} = R^{r}_{121}g_{r2} = R^{1}_{121}g_{12} + R^{2}_{121}g_{22} . \end{equation}

and all the Christoffel symbols are given by

\begin{equation} \begin{split} \Gamma_{11}^1 &= \frac{1}{2}g^{11}\left( \partial_1 g_{11} + \partial_1 g_{11} - \partial_1 g_{11} \right) + \frac{1}{2}g^{12}\left( \partial_1 g_{12} + \partial_1 g_{12} - \partial_2 g_{11} \right)\\ &= \frac{1}{2}g^{11}\left( \partial_1 g_{11} \right) + \frac{1}{2}g^{12}\left( 2\partial_1 g_{12} - \partial_2 g_{11} \right) \end{split} \end{equation} \begin{equation} \begin{split} \Gamma_{11}^2 &= \frac{1}{2}g^{21}\left( \partial_1 g_{11} + \partial_1 g_{11} - \partial_1 g_{11} \right) + \frac{1}{2}g^{22}\left( \partial_1 g_{12} + \partial_1 g_{12} - \partial_2 g_{11} \right)\\ &= \frac{1}{2}g^{21}\left( \partial_1 g_{11} \right) + \frac{1}{2}g^{22}\left( 2\partial_1 g_{12} - \partial_2 g_{11} \right) \end{split} \end{equation} \begin{equation} \begin{split} \Gamma_{12}^1 &= \frac{1}{2}g^{11}\left( \partial_1 g_{21} + \partial_2 g_{11} - \partial_1 g_{12} \right) + \frac{1}{2}g^{12}\left( \partial_1 g_{22} + \partial_2 g_{12} - \partial_2 g_{12} \right)\\ &= \frac{1}{2}g^{11}\left( \partial_2 g_{11} \right) + \frac{1}{2}g^{12}\left( \partial_1 g_{22} \right) \end{split} \end{equation} \begin{equation} \begin{split} \Gamma_{12}^2 &= \frac{1}{2}g^{21}\left( \partial_1 g_{21} + \partial_2 g_{11} - \partial_1 g_{12} \right) + \frac{1}{2}g^{22}\left( \partial_1 g_{22} + \partial_2 g_{12} - \partial_2 g_{12} \right)\\ &= \frac{1}{2}g^{21}\left( \partial_2 g_{11} \right) + \frac{1}{2}g^{22}\left( \partial_1 g_{22} \right) \end{split} \end{equation} \begin{equation} \begin{split} \Gamma_{22}^1 &= \frac{1}{2}g^{11}\left( \partial_2 g_{21} + \partial_2 g_{21} - \partial_1 g_{22} \right) + \frac{1}{2}g^{12}\left( \partial_2 g_{22} + \partial_2 g_{22} - \partial_2 g_{22} \right)\\ &= \frac{1}{2}g^{11}\left(2\partial_2 g_{21} - \partial_1 g_{22} \right) + \frac{1}{2}g^{12}\left( \partial_2 g_{22} \right) \end{split} \end{equation} \begin{equation} \begin{split} \Gamma_{22}^2 &= \frac{1}{2}g^{21}\left( \partial_2 g_{21} + \partial_2 g_{21} - \partial_1 g_{22} \right) + \frac{1}{2}g^{22}\left( \partial_2 g_{22} + \partial_2 g_{22} - \partial_2 g_{22} \right)\\ &= \frac{1}{2}g^{21}\left(2\partial_2 g_{21} - \partial_1 g_{22} \right) + \frac{1}{2}g^{22}\left( \partial_2 g_{22} \right) \end{split} \end{equation}

Consider the upper half plane

\begin{equation} H = \left\{ (x, y) \in \mathbb{R}^2 \, | \, y >0 \right\}, \end{equation}

together with the following metric

\begin{equation} ds^2 = \frac{dx^2 + dy^2}{y^2}. \end{equation}

Show that the Gauss curvature \(K\) is constant and has a negative value.

A vector \(\vec{v}\) with components \(v_1, v_2, v_n\) at a point \(p\) in Euclidean space \(\mathbb{R}^n\) can be thought of as an operator on differentiable functions as follows: If \(f\) is a differentiable function on a neighbourhood around \(p\), then \(\vec{v}\) assigns the real number \(\vec{v}(f)\) given by the directional derivative:

\begin{equation} \vec{v}(f) = v_1 \frac{\partial f (p)}{\partial x_1} + \cdots + v_n \frac{\partial f (p)}{\partial x_n}. \end{equation}

This operation of the vector \(\vec{v}\) on differentiable functions satisfies two important properties:

• \(\vec{v}(f + a g) = \vec{v}(f) + a\vec{v}(g)\). (\(\vec{v}\) acst linearly on differentiable functions around \(p\)).
• \(\vec{v}(f g) = \vec{v}(f) g(p) + f(p) \vec{v}(g)\). (\(\vec{v}\) satisfies the Leibniz rule, a.k.a derivation).

This motivates another definition of tangent vectors suitable for (abstract) manifolds. They will be directional derivatives, that is, linear derivations on functions. The collection of tangent vectors on a manifold \(M\) at a point \(p\in M\), will be also denoted by \(T_pM\). It is also a real vector space

In the following, wen we write manifold, we mean a smooth manifold. This class includes the regular surfaces \(S\in \mathbb{R}^3\) that we have treated earlier.

Remark. Let \(M\) be a smooth manifold, \(p \in M\) and \((U, \varphi)\) be a chart around \(p\). Then \(\partial_j|_{p}\) is the tangent vector that takes a function \(f:U\subset M\to \mathbb{R}\), given by

\begin{equation} \left. \partial_j \right|_{p} (f) := \left.\frac{\partial(f \circ \varphi^{-1})}{\partial x_j} \right|_{\varphi(p)}. \end{equation}

Let \(T_{p}M^*\otimes T_{p}M^*\) denote the set of bilinear maps on \(T_{p}M\). And consider the disjoint union

\begin{equation} TM^* \otimes TM^*:= \bigsqcup_{p\in M} T_{p}M^*\otimes T_{p}M^*. \end{equation}

Example.

The natural inner product \(\langle \cdot, \cdot \rangle\), in \(\mathbb{R}^3\) induces an inner product on each tangent space (plane) \(T_pS\). We choose to call this mapping \(g\) such that for every \(p\in S\), \(g(p) := g_p\) is an inner product on \(T_{p}S\).

Then \(g_p: T_{p}S \times T_{p}S \to \mathbb{R}\) is bilinear, symmetric and positive definite, more specifically, for any \(\vec{v},\vec{w},\vec{z}\in T_{p}S\), and \(a\in \mathbb{R}\)

• \(g_p(a\vec{v} +\vec{w}, \vec{z}) = ag_p(\vec{v}, \vec{z}) + g_{p}(\vec{w},\vec{z})\), and \(g_p(\vec{v}, a\vec{w} + \vec{z}) = ag_p(\vec{v}, \vec{w}) + g_{p}(\vec{v},\vec{z})\).
• \(g_p(\vec{v},\vec{w}) = g_p(\vec{w},\vec{v})\).
• \(g_{p}(\vec{v},\vec{v}) \geq 0\) and \(g_{p}(\vec{v},\vec{v})=0\) if and only if \(\vec{v}=0\).

To produce higher degree tensors, we define the tensor product \(\otimes\). Then if we have two 1 forms \(v,w\), their tensor product is the mapping \(v\otimes w: \mathbb{R}^n \to (\mathbb{R}^n)^*\times (\mathbb{R}^n)^*\), i.e., for every point \(p\in \mathbb{R}^n\), the map \((v\otimes w)_p: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\) is bilinear and defined as: \[ (v \otimes w)_p (A, B) = v_p(A) \, w_p(B). \]

If a 2-tensor \(\eta\) is such that \(\eta(A,B) = \eta(B,A)\), then the tensor is called symmetric.

If a 2-tensor \(\eta\) is such that \(\eta(A,A) \geq 0\) for all \(A\in \mathbb{R}^n\), and zero only for \(A=0\), then the tensor is called positive definite.

A special kind of 2-tensors are those that are anti-symmetric or alternating: \[ \eta(V,W) = - \eta(W,V). \]

Then, 2-forms are mappings \(\eta : \mathbb{R}^3 \to \Lambda((\mathbb{R}^3)^*\times(\mathbb{R}^3)^*)\) where \(\Lambda((\mathbb{R}^3)^*\times(\mathbb{R}^3)^*)\) is the set of all alternating bilinear maps.

From 1-tensors one can create 2-forms using the tensor product and alternating accordingly. Let \(dx,dy,dz\) the usual 1-forms in \(\mathbb{R}^3\). Then the wedge product \(dx \wedge dy\) at the point \(p\in \mathbb{R}^3\) defined by \[ (dx|_p\wedge dy|_p)(V_1,V_2) = (dx|_p \otimes dy|_p)(V_1,V_2) - (dx|_p \otimes dy|_p)(V_2,V_1), \] is an alternating bilinear form.

Since the wedge product is an algebraic definitions on each tangent space we have

\begin{equation} dx_{i} \wedge dx_{j} = \left( dx_{i} \otimes dx_{j} \right) - \left( dx_{j} \otimes dx_{1} \right), \end{equation}

where \[ dx_i|_p(\partial_j|_p) = \delta_{ij}. \]

A Riemannian metric on a smooth manifold \(M\) is a 2-tensor field \(g\) that is

• symmetric
• positive definite

A Riemannian metric thus determines an inner product on each tangent space \(T_p M\), which is typically written

\begin{equation} g_p(\vec{v}, \vec{w} ), \quad \mbox{for} \, \vec{v}, \vec{w} \in T_p M. \end{equation}

This inner product induces a norm on each tangent space \(T_p M\) and which will be denoted by

\begin{equation} \, |\vec{v}|_{g_{p}} = g_{p}(\vec{v}, \vec{v}). \end{equation}

Remark Clearly if \(S_1\) and \(S_2\) are globally isometric then they are locally isometric. But if \(S_1\) and \(S_2\) are locally isometric, then, it does not necessarily happens that they are globally isometric. The standard example is the cylinder and the plane.

The notion of isometry is the natural concept of equivalence for metric properties of regular surfaces, or more general, Riemannian manifolds.

In other words an Isometry establishes an equivalence between to geometric objects from the metric viewpoint.

Example A catenoid is locally isometric to the helicoid

1. The Identity map is an Isometry.
2. The Plane and the Cylinder in \(\mathbb{R}^3\), are locally isometric, but not globally isometric.

From the geometric definition of the standard inner product we see

\begin{equation} \cos{\theta_2} = \frac{\tilde{g}_{f(p)}\left( df_{p}(\vec{v}_1),df_{p}(\vec{v}_2)\right)}{|df_{p}(\vec{v}_1)|_{\tilde{g}_{f(p)}} |df_{p}(\vec{v}_2)|_{\tilde{g}_{f(p)}}} = \frac{\lambda^2(p) g_{p}\left( \vec{v}_1, \vec{v}_2\right)}{\lambda^2(p)|\vec{v}_1|_{g_p} |\vec{v}_2|_{g_p}} = \cos{\theta_1}. \end{equation}

• Which metrics \(g\) can be defined in \(\mathbb{S}^2\)?
• Which Riemannian manifolds \(\left(\mathbb{S}^2, g \right)\) may be embedded into \(\mathbb{R}^3\) isometrically?
• If \(\left(\mathbb{S}^2, g \right)\) has constant Gauss curvature equals to 1, then is this isometric to the standard sphere?
• The Weyl Problem and its solution by L. Nirenberg

The Euler characteristic is a topological invariant and denoted by \(\chi\). It was originally defined for polyhedra and used to prove various theorems about them including the classification of the Platonic solids according to the formula

\begin{equation} \chi = F - E + V, \end{equation}

where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron.

Solid	Vertices	Edges	Faces	\(\chi\)
Tetrahedron	4	6	4	2
Cube	8	12	6	2
Octahedron	6	12	8	2
Dodecahedron	20	30	12	2
Icosahedron	12	30	20	2

The Euler characteristic can be calculated easily for general surfaces by means of algebraic topology tools (that is, a description as a CW-complex) associating a polygon and using the above definitions.

Surface	\(\chi\)
Interval	1
Circle, \(\mathbb{S}\)	0
Disk \(D\)	1
Sphere, \(\mathbb{S}^2\)	2
Torus	0
Double Torus	-2
g-genus, closed orientable surface	2 - 2g

1. If \(\alpha:I \to S\) and \(\beta:J \to S\) are two regular parameterisation of \(\alpha(I)\) then \(w(t)\), \(t\in I\) is parallel if and only if \(w(\sigma)\), \(\sigma \in J\) is parallel, and in fact it holds \[ \frac{Dw}{d\sigma} = \frac{Dw}{dt}\frac{dt}{d\sigma}, \quad t\in I, \sigma \in J.\]
2. Consider any two fixed points \(p,q\in S\) and a parameterised curve \(\alpha:I \to S\) with \(\alpha(0) = p\) and \(\alpha(1) = q\). The map \(P_{\alpha}:T_pS \to T_qS\) which assigns to each \(v_p\in T_pS\) its parallel transport along \(\alpha\) at \(q\), is such that \(P_{\alpha}\) is an isometry.
3. If two surfaces \(S\) and \(\bar{S}\) are tangent along a parameterised curve \(\alpha\) and \(w_0\) is a vector of \(T_{\alpha(t_0)}S = T_{\alpha(t_0)}\bar{S}\), then \(w(t)\) is the parallel transport of \(w_0\) relative to the surface \(S\) if and only if \(w(t)\) is the parallel transport of \(w_0\) relative to \(\bar{S}\).

From now, on we will be assuming that the surface \(S\subset \mathbb{R}^3\) is orientable.

Remark. The regular connected curve \(C\) is geodesic if its parameterisation by arc-length is such that the acceleration vector \(\alpha''(s) = \kappa N\) is normal to the tangent plane. Then, we can say that a regular curve \(C \subset S\) with \(\kappa \neq 0\) is a geodesic if and only if its principal normal at each point \(p\in C\) is parallel to the normal to \(S\) at \(p\).

Remark Note that the sign of \(\left[ \frac{Dw(t)}{dt}\right]\) depends on the orientation of \(S\) and that one can compute its value by

\begin{equation} \left[ \frac{Dw(t)}{dt}\right] = \left\langle \frac{Dw(t)}{dt}, \nu \times w(t) \right\rangle. \end{equation}

Remark Note that the geodesics which are regular curves are then characterised as curves with zero geodesic curvature.

From the regularity of \(\alpha\), at each vertex \(\alpha(t_i)\) the following two limits are defined:

\begin{equation} \alpha'(t_i)^{-} = \lim_{t \to t_{i}} \alpha'(t), \quad t < t_{i} \end{equation}

and

\begin{equation} \alpha'(t_i)^{+} = \lim_{t \to t_{i}} \alpha'(t), \quad t > t_{i}. \end{equation}

In an orientable surface \(S\) let \(-\pi < \theta_i \leq \pi\) determined as follows, if the smallest (positive) angle between \(\alpha'(t_i)^{-}\) and \(\alpha'(t_i)^{+}\) is not \(\pi\), then we take this value and we give the same sign of the determinant \(\det(\alpha'(t_i)^{-},\alpha'(t_i)^{ + },\nu)\). In other words, the sign of \(\theta_i\) is given by the orientation of \(S\) (by the right hand rule). If the angle is \(\pi\), then the sign is determined by \(\det(\alpha'(t_i - \epsilon)^{-},\alpha'(t_i + \epsilon)^{ + },\nu)\), for a sufficiently small \(\epsilon >0\). This signed angle \(\theta_i\) is called the external angle at the vertex \(\alpha(t_i)\).

This is a topological result.

Let \(S \subset \mathbb{R}^3\) be an oriented regular surface. The subset \(R \subset S\) is called region if it is the union of connected open sets. The region \(R\) is called simple region if \(R\) is homeomorphic to a disk in \(\mathbb{R}^2\), and the boundary \(\partial R\) is the trace of a simple, closed, piecewise regular parameterised curve \(\alpha: I \to S\).

Let \(\alpha: I \to S\) be a simple, closed, piecewise regular parameterised curve. The tangent vector \(\alpha'(t)\) is a vector in the 2-dimensional vector space \(T_{\alpha(t)} S\). Hence there is a unique vector \(h(t)\in T_{\alpha(t)} S\) such that \(\{ \alpha'(t), h(t)\}\) is an orthogonal basis with a positive orientation. Then \(\alpha\) is called positively oriented if \(h(t)\) points towards \(R\) or equivalently, if for any other curve \(\beta: I \to \bar{R}\) with \(\beta(0) = \alpha(t)\) and \(\beta'(0) \neq \alpha'(t)\) then

\begin{equation} \langle \beta'(0) , h(t) \rangle > 0. \end{equation}

Let \(\varphi: U\subset \mathbb{R}^3 \to S\) be a parameterisation compatible with the orientation and let \(R \subset \varphi(U)\) be a bounded region of \(S\). If \(f\) is a differentiable function on \(S\), then we define the integral of \(f\) over the region \(R\) as follows

\begin{equation} \int_{R} f dS = \int\int_{\varphi^{-1}(R)} f(u,v) \sqrt{EG - F^2} du\, dv. \end{equation}

Remark Note that if \(C\) is a regular oriented curve on the surface \(S\), \(\alpha(s)\) a parameterisation by arc-length \(s\) of \(C\) passing through \(p\in C\), and let \(v(s)\) be a parallel field along \(\alpha(s)\). Take \(w(t) = \alpha'(s)\). Then we have

\begin{equation} \kappa_{g}(s) = \left[ \frac{D\alpha'(s)}{ds}\right] = \frac{d\rho}{ds}, \end{equation}

Equivalently

the geodesic curvature is the rate of change of the angle that the tangent to the curve makes with respect to a parallel direction along the curve.

Proof

Let \(\varphi : U\to S\) a local parameterisation of the oriented regular surface \(S\). We use coordinates \(\varphi = \varphi(u,v)\). Then we also write \(\alpha(t) = \varphi\left(u(t), v(t)\right)\). Using the previous result \eqref{eq:forGBT1} we can write:

\begin{equation} \kappa_g(s) = \frac{1}{2\sqrt{EG}} \left\{ G_{u}\frac{dv}{ds} - E_{v}\frac{du}{ds}\right\} + \frac{d \rho_i}{ds}, \end{equation}

where \(\rho_{i}(s)\) is the differentiable function measuring the positive angle between \(\partial_1 = \varphi_{u}\), and the velocity vector \(\alpha'(s)\) in \([s_{i}, s_{i+1}]\).

Integrating we obtain

\begin{equation} \sum_{i=0}^{k} \int_{s_{i}}^{s_{i+1}} \kappa_g(s)\, ds= \sum_{i=0}^{k} \int_{s_{i}}^{s_{i+1}} \left(\frac{1}{2\sqrt{EG}} \left\{ G_{u}\frac{dv}{ds} - E_{v}\frac{du}{ds}\right\}\right)\,ds + \sum_{i=0}^{k} \int_{s_{i}}^{s_{i+1}} \frac{d \rho_i}{ds}\, ds. \end{equation}

Now apply the Green's Theorem (Stokes theorem on the plane)

\begin{equation} \sum_{i=0}^{k} \int_{s_{i}}^{s_{i+1}} \left\{ P\frac{du}{ds} + Q\frac{dv}{ds}\right\}\,ds = \sum_{i=0}^{k} \int\int_{A} \left\{ \frac{\partial Q}{\partial u} - \frac{\partial P}{\partial v}\right\}\,du\,dv. \end{equation}

Then we obtain

\begin{equation} \sum_{i=0}^{k} \int_{s_{i}}^{s_{i+1}} \kappa_g(s)\, ds= \int \int_{\varphi^{-1}\{R\}} \left\{\left(\frac{E_{v}}{2\sqrt{EG}}\right)_{v} + \left(\frac{G_{u}}{2\sqrt{EG}}\right)_{u}\right\} \,du\,dv + \sum_{i=0}^{k} \int_{s_{i}}^{s_{i+1}} \frac{d \rho_i}{ds}\, ds. \end{equation}

The term involving the double integral is

\begin{equation} \begin{split} \int \int_{\varphi^{-1}\{R\}} \left\{\left(\frac{E_{v}}{2\sqrt{EG}}\right)_{v} + \left(\frac{G_{u}}{2\sqrt{EG}}\right)_{u}\right\} \,du\,dv & = -\int \int_{\varphi^{-1}\{R\}} K \sqrt{EG}\,du\,dv \\ &=-\int_{R} K dS. \end{split} \end{equation}

From the turning tangent theorem \eqref{eq:TTT}, we have the following identity

\begin{equation} \sum_{i=0}^{k} \int_{s_{i}}^{s_{i+1}} \frac{d \rho_i}{ds}\, ds =\sum_{i=0}^{k} \left(\rho_{i}(s_{i+1}) - \rho_{i}(s_{i})\right) = \pm 2\pi - \sum_{i=0}^{k} \theta_i. \end{equation}

Use the fact that \(\alpha\) is positively oriented and conclude that one should choose the positive sign.

It follows finally that

\begin{equation} \sum_{i=0}^{k} \int_{s_{i}}^{s_{i+1}} \kappa_{g}(s) ds + \int_{R} K dS + \sum_{i=0}^{k}\theta_i = 2\pi. \end{equation}

As before, consider

• \(\varphi:U\to S\) a local orthogonal parameterisation.
• \(R\subset \varphi(U)\) a simple region without vertices containing \(p\in S\).
• \(\alpha: [0,l] \to \varphi(U)\) a curve parameterised by arc length, with \(\alpha(I) = \partial R\).

Let \(W_0\) be a unit vector tangent to \(S\) at \(\alpha(0)\), and its parallel transport along the curve be denote by \(W(s)\). Then we have the following identities

\begin{equation} 0 = \left[\frac{DW}{ds}\right] = \frac{1}{2\sqrt{EG}} \left\{ G_{u}\frac{dv}{ds} - E_{v}\frac{du}{ds}\right\} + \frac{d\rho}{ds}, \end{equation}

Then integrating on both sides we obtain

\begin{equation} 0 = \int_{0}^{l} \frac{1}{2\sqrt{EG}} \left\{ G_{u}\frac{dv}{ds} - E_{v}\frac{du}{ds}\right\} ds + \int_{0}^{l}\frac{d\rho}{ds} = - \int\int_{R} KdS + \rho(l) - \rho(0), \end{equation}

hence

\begin{equation} \lim_{R \to p} \frac{\Delta \rho}{A(R)} = K(p). \end{equation}

Remark Notes that the global Gauss-Bonnet theorem implies the Gauss-Bonnet theorem for simple connected regions.