By 苏剑林 | Nov 04, 2016
Preface
In the "Understanding Riemannian Geometry" series, I shared some "geometric" insights into Riemannian geometry, while leaving one question behind: how do we actually calculate the Riemann tensor? MTW's Gravitation mentions a method based on exterior calculus, but I was unfamiliar with exterior calculus, so I set out to learn it. Indeed, it was the efficient steps for calculating the curvature tensor in Gravitation that made me decide to delve deeper into exterior calculus. As it turns out, efficiency is a powerful primary motivator.
This series of articles mainly shares some insights from learning exterior calculus. It has undergone multiple revisions and improvements and covers a wide range of content, such as exterior products, moving frames, exterior derivatives, and their applications in Riemannian geometry, finally concluding with an effective way to calculate curvature.
Notation Notes: In this series, bold letters represent vectors, matrices, and bases; ordinary letters represent scalars, which may be scalar functions or vector components. Unless otherwise specified, $n$ represents the dimension of the space (manifold). Einstein summation convention is used throughout, where identical upper and lower indices imply summation from $1$ to $n$, i.e., $\alpha_{\mu}\beta^{\mu}=\sum_{\mu=1}^{n} \alpha_{\mu}\beta^{\mu}$. Conventionally, the subscript is written first, for example, $\alpha_{\mu}\beta^{\mu}$ is effectively the same as $\beta^{\mu}\alpha_{\mu}$, but it is standard practice to write the former. Common notations include: $\mu, \nu$ for component indices, $x^{\mu}$ for coordinate components, $dx^{\mu}$ for components of tangent vectors (differentials), and Greek letters like $\alpha, \beta, \omega$ for differential forms. Symbols may be reused, but their meanings are typically explained near their appearance, so there should be no confusion.
Finally, I should mention that I don't yet have a particularly strong intuition for exterior calculus, so there may be errors in the articles. I hope readers will forgive them and point them out. This series is titled "An Introduction to Exterior Calculus" (literally "Shallow Talk"); it is not out of modesty—my understanding really is shallow, and my explanations are quite basic~
Vector Inspiration
Since we first learned the concept of vectors in high school, vectors have involved two methods of operation. One is to establish a coordinate system and use coordinates for computation; the other is to operate directly using vector rules. To deal with solid geometry in college entrance exams, students tend to practice coordinate-based calculations more—the language of components. However, component language sometimes hinders our understanding of vectors as "objective entities," and component language is not always the simplest approach. For example, consider the following simple problem:
Let $\boldsymbol{A}, \boldsymbol{B}$ be two vectors of equal magnitude. Prove that $\boldsymbol{A}-\boldsymbol{B}$ is perpendicular to $\boldsymbol{A}+\boldsymbol{B}$.
The standard answer to this question should be $(\boldsymbol{A}-\boldsymbol{B})\cdot (\boldsymbol{A}+\boldsymbol{B}) = \boldsymbol{A}^2 - \boldsymbol{B}^2 = 0$, thus proving that $\boldsymbol{A}-\boldsymbol{B}$ is perpendicular to $\boldsymbol{A}+\boldsymbol{B}$. I believe nobody would want to establish a coordinate system, specify every component, and then compute. Moreover, this conclusion holds for any $n$-dimensional space, whereas once a coordinate system is established, it implies fixing the dimension of the space, specializing a general conclusion. That is to say, there exists a pure vector formal language that has a unique advantage in describing and operating on vectors as "objective entities."
Note: In this article and the "Understanding Riemannian Geometry" series, from the perspective of tensor language, vectors are all contravariant vectors. In fact, I believe the concepts of covariance and contravariance are unnecessary; "true" vectors are all contravariant, while so-called covariant vectors are purely defined. From a geometric perspective, one can successfully complete all necessary tasks without the concepts of contravariance and covariance. We don't need to know whether a quantity is covariant or contravariant; we only need to judge whether it is a geometric quantity. This approach might make deriving certain algebraic expressions more difficult, but the conceptual understanding is much deeper.