Excitingly, the path we took to derive the Riemann curvature also allows us to catch a glimpse of the Gauss–Bonnet formula, providing a true experience of the flavor of intrinsic geometry research.
The Gauss–Bonnet formula is a classic formula in global differential geometry that establishes a connection between the local properties and the global properties of a space. Starting from a geometric path and combining elements of matrix transformations and mathematical analysis, we have progressively derived geodesics, covariant derivatives, and the curvature tensor. Now, we can also obtain the classic Gauss–Bonnet formula, showing that we have traveled quite far along this path. Although the process is not perfect, it has not deviated from the core of this series: geometric intuition. The purpose of this article is to share an intuitive way of thinking about Riemannian geometry. Since it is about ideas, the focus is on the exchange of thoughts rather than rigorous proof. Therefore, for everyone, this series should be regarded as supplementary material for Riemannian geometry.
Rewriting the Form
First, we can rewrite Equation (48) into a form with more geometric significance. Starting from \Delta A^{\mu} = -R^{\mu}_{\alpha\beta\gamma} A^{\alpha} dx^{\beta}\delta x^{\gamma} = -g^{\mu\nu}R_{\nu\alpha\beta\gamma} A^{\alpha} dx^{\beta}\delta x^{\gamma} \tag{54} we swap the positions of \beta and \gamma to get \Delta A^{\mu} = -g^{\mu\nu}R_{\nu\alpha\gamma\beta} A^{\alpha} dx^{\gamma}\delta x^{\beta} \tag{55} Using the property R_{\nu\alpha\beta\gamma} = -R_{\nu\alpha\gamma\beta} and adding the two equations, we obtain \begin{aligned} \Delta A^{\mu} &= -\frac{1}{2}g^{\mu\nu}R_{\nu\alpha\beta\gamma} A^{\alpha} (dx^{\beta}\delta x^{\gamma} - dx^{\gamma}\delta x^{\beta}) \\ &= -\frac{1}{2}g^{\mu\nu}R_{\nu\alpha\beta\gamma} A^{\alpha} \det\begin{pmatrix} dx^{\beta} & \delta x^{\beta} \\ dx^{\gamma} & \delta x^{\gamma} \end{pmatrix} \end{aligned} \tag{56} The general geometric meaning of this expression requires interpretation using exterior calculus and surface integrals, which we will not discuss. However, we can consider the special case of a two-dimensional space (i.e., considering a two-dimensional surface in three-dimensional Euclidean space), where the geometric meaning becomes clear. When n=2, each summation index actually only has two terms. That is: \Delta A^{\mu} = -\frac{1}{2}\sum_{\nu=1}^2 \sum_{\alpha=1}^2 \sum_{\beta=1}^2 \sum_{\gamma=1}^2 g^{\mu\nu}R_{\nu\alpha\beta\gamma} A^{\alpha} \det\begin{pmatrix} dx^{\beta} & \delta x^{\beta} \\ dx^{\gamma} & \delta x^{\gamma} \end{pmatrix} \tag{57} We can first calculate the summation over \beta and \gamma. Because of the determinant, it is only meaningful when \beta \neq \gamma. Then, utilizing R_{\nu\alpha\beta\gamma} = -R_{\nu\alpha\gamma\beta}, we get \Delta A^{\mu} = -\sum_{\nu=1}^2 \sum_{\alpha=1}^2 g^{\mu\nu}R_{\nu\alpha 12} A^{\alpha} \det\begin{pmatrix} dx^{1} & \delta x^{1} \\ dx^{2} & \delta x^{2} \end{pmatrix} \tag{58} Next, consider the summation over \nu and \alpha. Similarly, it is only meaningful when \nu \neq \alpha, and we also have the antisymmetry R_{\nu\alpha\beta\gamma} = -R_{\alpha\nu\beta\gamma}. Therefore, we can obtain \Delta A^{\mu} = - (g^{\mu 1} A^{2} - g^{\mu 2} A^{1}) R_{12 12} \det\begin{pmatrix} dx^{1} & \delta x^{1} \\ dx^{2} & \delta x^{2} \end{pmatrix} \tag{59} This can be rewritten as \Delta A^{\mu} = - \sqrt{g}(g^{\mu 1} A^{2} - g^{\mu 2} A^{1}) \frac{R_{12 12}}{g} \sqrt{g} \det\begin{pmatrix} dx^{1} & \delta x^{1} \\ dx^{2} & \delta x^{2} \end{pmatrix} \tag{60} Note that in two-dimensional space, \sqrt{g} \det\begin{pmatrix} dx^{1} & \delta x^{1} \\ dx^{2} & \delta x^{2} \end{pmatrix} has a clear geometric meaning: it is the area of the parallelogram spanned by the vector (dx^1, dx^2) and the vector (\delta x^1, \delta x^2) (refer to the result of Equation (15)), which we simply denote as \Delta S. Furthermore, \frac{R_{12 12}}{g} is exactly the Gaussian curvature K as defined in differential geometry. Thus, it can be written as \Delta A^{\mu} = - \sqrt{g}(g^{\mu 1} A^{2} - g^{\mu 2} A^{1}) K \Delta S \tag{61}
Angular Difference Change
Now, we can analyze the angle between the vector A^{\mu} and the vector after it has changed to A^{\mu} + \Delta A^{\mu}. Assuming A^{\mu} is a unit vector, we first need to calculate the inner product: \begin{aligned} &\frac{g_{\mu\nu}A^{\mu}(A^{\nu} + \Delta A^{\nu})}{\sqrt{g_{\mu\nu}A^{\mu}A^{\nu}}\sqrt{g_{\mu\nu}(A^{\mu} + \Delta A^{\mu})(A^{\nu} + \Delta A^{\nu})}} \\ =& \frac{1 + g_{\mu\nu}A^{\mu}\Delta A^{\nu}}{\sqrt{1 + 2g_{\mu\nu}A^{\mu}\Delta A^{\nu} + g_{\mu\nu}\Delta A^{\mu}\Delta A^{\nu}}} \end{aligned} \tag{62} Since \cos \Delta \theta = 1 - \frac{\Delta\theta^2}{2} + \dots, we need to calculate up to the second-order terms, namely the \Delta A^{\mu} \Delta A^{\nu} term. Approximating to the second order, the result is 1 - \frac{1}{2}\left[g_{\mu\nu}\Delta A^{\mu}\Delta A^{\nu} - (g_{\mu\nu}A^{\mu}\Delta A^{\nu})^2 \right] \tag{63} So \begin{aligned} \Delta \theta &= \sqrt{g_{\mu\nu}\Delta A^{\mu}\Delta A^{\nu} - (g_{\mu\nu}A^{\mu}\Delta A^{\nu})^2} \\ &= \sqrt{(g_{\mu\nu} A^{\mu} A^{\nu})(g_{\mu\nu}\Delta A^{\mu}\Delta A^{\nu}) - (g_{\mu\nu}A^{\mu}\Delta A^{\nu})^2} \end{aligned} \tag{64} This is actually the area of the parallelogram spanned by A^{\mu} and \Delta A^{\mu}. When n=2, this is \sqrt{g} \det\begin{pmatrix} A^1 & \Delta A^1 \\ A^2 & \Delta A^2 \end{pmatrix} = \sqrt{g} (A^{1}\Delta A^{2} - A^{2}\Delta A^{1}). Substituting the expressions for \Delta A^{1} and \Delta A^{2}, we get \Delta \theta = g\left[g^{22}(A^1)^2 - g^{12}A^2 A^1 - g^{21}A^1 A^2 + g^{11}(A^2)^2\right] K \Delta S \tag{65} For a 2 \times 2 matrix, there is an inversion formula: \begin{aligned} \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix}^{-1} &= \frac{1}{g_{11}g_{22} - g_{12}g_{21}} \begin{pmatrix} g_{22} & -g_{12} \\ -g_{21} & g_{11} \end{pmatrix} \\ &= \frac{1}{g} \begin{pmatrix} g_{22} & -g_{12} \\ -g_{21} & g_{11} \end{pmatrix} \end{aligned} \tag{66} Therefore g^{11} = \frac{g_{22}}{g}, \quad g^{12} = -\frac{g_{12}}{g}, \quad g^{21} = -\frac{g_{21}}{g}, \quad g^{22} = \frac{g_{11}}{g} \tag{67} Substituting into Equation (65), we obtain \Delta \theta = [g_{11} (A^1)^2 + g_{12} A^1 A^2 + g_{21} A^2 A^1 + g_{22} (A^2)^2 ] K \Delta S = K \Delta S \tag{68} The last equality holds because we assumed from the beginning that A^{\mu} is a unit vector. Thus, the angular difference produced after a vector returns along a small closed curve is equal to the product of the Gaussian curvature K and the area element \Delta S. Consequently, we can deduce that if a vector is parallel transported back along a large-scale closed curve \mathbb{C}, the change is expressed as a surface integral: \Delta \theta = \int_{\mathbb{C}} K d S \tag{69} This is the main content of the Gauss–Bonnet formula in differential geometry, stating that the angular difference equals the surface integral of the Gaussian curvature. Topics such as the sum of the interior angles of a spherical triangle are related to it. It is one of the founding works of global differential geometry.
A Few Comments
It is worth mentioning that our discussion above is entirely intrinsic, meaning it does not introduce the concept of a surface in three-dimensional space, which is very appealing. The great mathematician Chern Shiing-shen once said that the best work of his life was the intrinsic proof of the higher-dimensional Gauss–Bonnet formula (proofs before him were extrinsic). It can be seen that purely intrinsic work is the pursuit of Riemannian geometry research. Of course, what we have here is at most an illustrative guide rather than a complete proof. But for this series, this level is sufficient.
Note that one point that might confuse readers is: as a volume (or area), it should be non-negative, but if written in the form of a matrix determinant, it can be positive or negative, which seems contradictory. This is indeed difficult to clarify without introducing exterior calculus. Within the scope of elementary analysis, the only solution is to take the absolute value if a negative volume appears.
When reprinting, please include the original address of this article: https://kexue.fm/archives/4033
For more detailed reprinting matters, please refer to: Scientific Space FAQ