The differential-forms tag has no wiki summary.
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Illustration of vector calculus vs. differential forms
I am looking for a nice illustration of how vector calculus relates to differential forms. A demonstration that employs physics is appreciable (e.g. electromagnetism).
In particular, while dualizing ...
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Two “different” adjoints of exterior derivative on manifolds with boundary in the $L^2$-setting
The follow problem appears in the setting of $L^2$-differential forms on manifolds with boundary. An abstracted operator theoretic problem is given below.
Suppose $M$ is a smooth Riemannian manifold ...
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1answer
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Bundle orientability vs manifold orientability
Given a vector bundle, I am a bit hazy about the difference between the notions of its orientability as a bundle and as a manifold.
I think I know that the following are true,
A tangent bundle of a ...
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1answer
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Is there a Stokes theorem for covariant derivatives?
A $V$-valued differential form on $M$ is a smooth map $\omega : TM \to V$ such that $\omega$ restricted to any tangent space $T_p M$ is an element of the $V$-valued exterior algebra $\Lambda^n (T_p M, ...
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1answer
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If $S^{2n+1}$ is covering space of $X$, then $X$ is orientable.
Is there any direct way to prove that $n$-manifold is orientable? In AT we can just calculate $n$'th homology group and check whether it's $\mathbb Z$ or $0$. But I want a geometric method, using ...
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Geometric interpretation of connection forms, torsion forms, curvature forms, etc
I have just begun learning about the connection 1-forms, torsion 2-forms, and curvature 2-forms in the context of Riemannian manifolds. However, I am finding it hard to relate these notions to any ...
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The canonical (1,1)-form on a compact Riemann surface gives locally a subharmonic function
Let $X$ be a compact connected Riemann surface of genus $g>0$.
We have the so called canonical (1,1)-form $\mu$ on $X$ defined as follows. Choose an orthonormal basis $(\omega_1,\ldots, \omega_g)$ ...
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functoriality of derivations
I seem to have problems understanding algebraically why given a map of manifolds $f: M \to N$ we get a bundle map $TM \to f^*TN$.
Now, fiberwise it's all good. But I do not understand how to define ...
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1answer
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Exercise concerning the Lefschetz fixed point number
I can't see a good approach to the third part of the following problem:
Let $f: M \to M$ be a smooth map of a compact oriented manifold into itself. Denote by $H^q(f)$ the induced map on the ...
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2answers
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Examples of Computations in Algebraic Topology
I have started reading "Differential Forms in Algebraic Topology" by Bott, Tu, recently. While I'm quite happy with the exposition of the theorems and explanation of theoretical results, I'm missing ...
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1answer
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How to find or guess the homotopy operator?
In the proof of the Poincare Lemma for compactly supported cohomology,the homotopy operator K suddenly appears and satisfies the equation 1-e*π*=±（dK-Kd),that is too lucky!I do not know how to find or ...
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1answer
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How to prove that this kind of differential form exists on an algebraic curve?
The following is a problem in Miranda's Algebraic Curves and Riemann Surfaces.
Given any algebraic curve $X$ and a point $p \in X$, show that there is a meromorphic $1$-form $\omega$ on $X$ whose ...
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1answer
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Coordinate free proof that curvature is the “square” of the connection
Here's the setup. Consider a vector bundle $E$ over a manifold $M$ and let $\Omega^*(M, E)$ denote the space of $E$-valued differential forms (i.e. the space of sections of the vector bundle ...
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1answer
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Differential Form on a Riemann Surface
The following problem is basically from Miranda's "Algebraic Curves and Riemann Surfaces", which I am reading on my own; if there are any rules against posting textbook problems, my apologies!
Let ...
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2answers
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Intuition behind $dx \wedge dy=-dy \wedge dx$
I was re-reading this old book of mine; and I noticed that in defining the rules of differential forms, it "makes sense" that we have the rule $dx \wedge dx=0$ because if $dx$ is infinitesimal, then ...
