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What are „differential forms“

differential geometry
differential forms
tangent space
cotangent space
tangent bundle
cotangent bundle
exterior algebra
global section

Luca Leon Happel


May 27, 2022


Currently while visiting the “KoMa” (Konferenz der deutschsprachigen Mathematikfachschaften / conference of German-speaking mathematics students) I got into a lot of interesting conversations about geometry, topology and category theory. This set the stage for me to think about differential forms because I wanted to understand how exactly these things look and feel like.

My sketches

Notes about differential forms

The proof mentioned on page 1 is viewable on https://web.archive.org/web/20211224033113/https://planetmath.org/tensorproductofdualspacesisadualspaceoftensorproduct


Throughout this article we will choose the following symbols and notation:

  • \(M\) a smooth manifold over a field \(K\subset \mathbb{R}\)
  • \(p\in M\) a point in \(M\)
  • \(T_p M\) the tangent space of \(M\) at \(p\)
  • \(T_p^\ast M\) the cotangent space of \(M\) at \(p\), which consists of the linear functions \(\phi: T_p M \to K\). For finite tangent spaces \(T_p M\), we can canonically identify \(T_p^\ast M\) with \(T_p M\).
  • \(T M = \bigcup_{p\in M} \{p\}\times T_p M\) and \(T^\ast M = \bigcup_{p\in M} \{p\}\times T_p^\ast M\) are the tangent/cotangent bundles of \(M\)
  • \(\Lambda^k(T^\ast M)\) is the k-th exterior product of the cotangent bundle of \(M\). Its elements \(\mu_p\in\Lambda^k(T^\ast M)\) are called k-coblades and allow to measure volumes on some tangent space \(T_pM\) in \(k\)-dimensions.
  • \(\Gamma(M, \Lambda^k(T^\ast M)) = \Gamma(\Lambda^k(T^\ast M))\) is the space of global sections from the basespace \(M\) to the total space \(\Lambda^k(T^\ast M)\). We use this, because for an element \(\omega:M \to \Lambda^k(T^\ast M)\) in it, we can smoothly associate for each point \(p\) on \(M\) a k-coblade.

Formal definition

A differential form \(\omega\) is defined as an element in the space of global sections over the kth exterior algebra of the cotangent bundle over a smooth manifold \(M\). We denote this as:

\[\omega \in \Gamma(M, \Lambda^k(T^\ast M))\]

My interpretation

Basically, a differential 1-form can be visualized as appending a vector onto each point of our manifold. Because we work with a smooth section, these vectors need to vary “smoothly” from one point to another. This means, if we wiggle the point we look at on our manifold just a tiny bit, the vector we have related to this point will also wiggle just a tiny bit. Also, our corresponding vector cannot take any sharp turns if we do not move our point in sharp turns.