# Tagged Questions

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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### Computing the Chern-Simons invariant of SO(3)

I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and ...
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### A Way to make the following “proof” of the Hairy Ball Theorem rigorous?

I plan on giving a talk soon to undergraduates and I'd like to talk about the hairy ball theorem during the talk. I was trying to think of some sort of visually intuitive proof of this fact. (I ...
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### Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
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### Wanted: A purely algebraic proof of the Frobenius theorem on distributions

Is there a purely algebraic proof of the Frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of ($R$-valued) derivations of the algebra $R$ endowed ...
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### Characterization of the exterior derivative $d$

In the paper Natural Operations on Differential Forms, the author R. Palais shows that the exterior derivative $d$ is characterized as the unique "natural" linear map from $\Phi^p$ to $\Phi^{p+1}$ (...
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### On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
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### Curvature $0$ and involutive horizontal distributions

I am trying to check why curvature $0$ implies that the horizontal distribution is involutive. Let $\pi:P\to U$ be a principal $G:=GL_n$ bundle. Assume that $P$ is trivial and $\pi$ admits a section. ...
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### Complement of a foliation

I have an $n$-manifold $M$ which is foliated by leaves $F_\alpha$ of dimension $p$ and a path $\gamma:[0,1]\to M$. You can take without problems $\gamma$ to be injective. Is the following statement ...
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### What is a moduli space for a differential geometer?

A moduli space is a set that parametrizes objects with a fixed property and that is endowed with a particular structure. This should be an intuitive and general definition of what a moduli space is. ...
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### Does Stokes' Theorem hold on spaces with singular points?

I have come across the question whether Stokes' theorem holds also on orbifolds. Let us take the simple case of $T^2/Z_2$ with a one-form $A$, then the question becomes: For a region $\Gamma$ with ...
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### Different notions of isometry for Riemannian 2-manifolds

There are two notions of isometry between Riemannian 2-manifolds: a distance-preserving map $f$ with $d(x,y) = d(f(x),f(y))$ and a "metric-preserving" map $f$ with $I(x) = I(f(x))$ ($I(x)$ being ...
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### Torsion in two dimensions?

This question is about the notion of a connection with torsion in differential geometry, i.e., a connection that is not Levi-Civita. (It's not about the torsion of a curve in three dimensions.) ...
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### How to Characterize Gradient Vector Fields?

Let $V$ be a vector field on a smooth manifold $M$. Are there nice conditions under which there exists a (Riemannian) metric on $M$ such that $V$ is the gradient of some smooth function on $M$? ...
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### Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
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### De Rham Cohomology of $M \times \mathbb{S}^1$

Let $M$ be a closed (compact, without boundary) $m$-dimensional manifold. I want to prove that $H^{k+1}(M \times \mathbb{S}^1) = H^k(M) \oplus H^{k+1}(M)$. ($H^k$ is the $k$-th De Rham cohomology ...
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### “Natural” constructions of tensor fields from tensor fields on a manifold

This question begins is related to this question on physics.SE Uniqueness of Riemann Curvature Tensor, which asks roughly "what tensors can we make locally out of just the metric tensor? We can ...
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### What is the volume of Complex Projective Space with Fubini-Study Metric?

I try to compute the volume of the complex projective space $\mathbb{CP}^n$ with Fubini-Study metric, normalized to have diameter $=\pi/2$ i.e. the sectional curvatures lie between $1$ and $4$. Fix a ...
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### Does there exist a vector field s.t. all orbits are dense on $\mathbb{R}^2$

Is there a complete vector field such that the all orbits are dense on a contractible manifold? For example, $\mathbb{R}^2$,the interior of a $n$-unit ball.
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### Connection in fibre bundle from discontinuous group action

I am trying to understand connections in fibre bundles. I thought of the following problem: Let $\Gamma$ be the discrete group generated by \begin{pmatrix} 1 & 3 & 0 \\ 0 & 1 & 0 \\ ...
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### Trying to Understand Lefschetz Pencils

I'm reading on Lefschetz pencils, and I'm trying to understand the following condition ii) better, although I would appreciate insights on condition i), and in general. A Lefschetz pencil on a $4$...
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### What's the idea behind the covariant derivative?

I'm learning differential geometry from what I find on the Internet (to eventually find a grasp on General Relativity too). Right now I playing with a sphere. I have 3 functions ($x$, $y$, $z$) that ...
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### How are boundary conditions formally captured by the jet bundle approach to differential equations?

In the jet bundle approach to differential equations https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations one identifies the equation with the set of a solution of the ...
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### When is There a Solution to “Pullback Equation” of Differential Forms

All: Let $f: M \to N$ be a smooth map between manifolds, and let $w$ be a $1$-form on $M$. Under what conditions is there a $1$-form $z$ defined on $N$ so that $w=f^*z$, i.e., so that $w$ is the ...
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### Group actions and associated bundles

Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
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### Difference between parallel transport and derivative of the exponential map

Given a Riemannian manifold $M$, let $c(t) = \exp_p(tX)$ be the geodesic emanating from $p \in M$ with initial value $X$. Let $t_0$ be small enough, then we have to ways to map $T_pM$ to $T_{c(t_0)} M$...
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