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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

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Uniqueness of differentiable structure

I'm trying to solve the following question: Let $F:M\longrightarrow N$ be a bijective map. Prove that, if M is an $n$-dimensional differentiable manifold, then $N$ admits a unique differentiable structure making $F$ a diffeomorphism. I think that…
Wild Feather
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curve evolution length formula

I run to the following problem which says if you have a smooth curve that is evolving over time (say finite length at the beginning) then $$\frac{d}{dt}(curve \; length \; at \; time \; t)=-\int_{curve} k\cdot v \; ds,$$ where $k$ is curvature of…
dmm
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What is the curvature 2-form on the associated vector bundle?

Consider a principal bundle $P\rightarrow M$ and the associated vector bundle $P\times_{\rho}V$ over $M$ such that $(p,v)=(pg^{-1},\rho(g)v)$. The connection on the principal bundle $A$ defines a covariant derivative on $P\times_{\rho}V$ as follows.…
Bombyx mori
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$M$ is orientable $\Leftrightarrow$ determinant bundle $\det(TM)$ is trivial

Let $M$ be a differentiable manifold and $TM$ be its tangent bundle. I need to prove the following: $M$ is orientable if and only if $\det(TM)$ is trivial. The definition of determinant bundle I'm using is the following: Given a vector bundle $E$…
un umile appassionato
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If $F$ is diffeomorphism the linear map $F_*$ is an isomorphism?

Let $f\colon M \rightarrow N$ be a smooth map. If $f$ is a diffeomorphism I am trying to show that the linear map $f_*$ : $T_pM \rightarrow T_{f(p)}M$ is an isomorphism for all $p \in M$. I know the derivative map $T_pf\colon T_pM \rightarrow…
L.S.
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Definition of a distribution and integral manifolds

We are currently talking about the various forms of the Frobenius theorem in my differential geometry class in order to build up integrability. For one of the versions, we use distributions, and I'd like to get a few things straight. Given a…
Mobius
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Deciding whether a given set is a regular surface

I'm trying to decide whether the following set is a regular surface: $$\{(x^3 - 3xy^2, 3x^2y - y^3, 0) : (x, y) \in \mathbb{R}^2\}$$ I know that the map ${\bf x} : \mathbb{R^2} \to \mathbb{R^3}$ defined by ${\bf x}(x, y) = (x^3 - 3xy^2, 3x^2y -…
rt93
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Frames and Co-frames

I'm trying to write a precise definition of a frame and co-frame. I have that a frame is a basis for a tangent space $T_pM$ and the co-frame is the dual basis (basis for the co-tangent space $T^{\ast}_pM$) where $M$ is a manifold. In coordinates, I…
TheMobiusLoops
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Chern number require G-bundles?

Do we need a principle G-bundle to define a Chern number or is it enough to have a vector bundle? (and in the case this needs to have complex fibers?) That is, is it mandatory to consider the action on the fibers of a Lie group? I'm asking this…
ShortEdge
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Directional derivative along a curve (Covariant derivatives)

I am having a hard time understanding covariant derivatives. My main problem is working with concrete example. So I was wondering if anybody could help me with explaining it by using simple example. Let us say we have the…
Novo
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Check that a curve is a geodesic.

Suppose $M$ is a two-dimensional manifold. Let $\sigma:M \rightarrow M$ be an isometry such that $\sigma^2=1$. Suppose that the fixed point set $\gamma=\{x \in M| \sigma(x)=x\}$ is a connected one-dimensional submanifold of $M$. The question asks to…
Honghao
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Construction of cut-off function

I want to know the example of cut-off function. $\phi \in C^2 ([0,\infty))$ satisfies the followings : (1) $\phi(x) =1$ on $[0,r]$ (2) $\phi(x) =0$ on $ x > 2r$ (3) $- C r^{-1} \phi^{1/2}(x) \leq \phi ' (x) \leq 0$ on $r \leq x \leq 2r $ (4) $|…
HK Lee
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What is the geodesic equation on $\mathbb{S}^{n}$?

Suppose $\gamma: \mathbb{R}\rightarrow \mathbb{S}^{n}$ is a smooth curve. Let $\gamma(t)=(x^{1}(t)...x^{n+1}(t))$. Let $\mathbb{D}^{n}$ be embedded into $\mathbb{R}^{n+1}$ by viewing $\mathbb{R}^{n+1}$ as $\mathbb{R}^{n}\times \mathbb{R}$ and…
Bombyx mori
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Using a bicycle to calculate areas

I'm organizing a maths divulgation thing on my school and I found this interesting talk on youtube where a mathematician uses a bicycle to calculate the area inside a closed curve. Here it is: https://youtu.be/hukIyIYjto4?t=2827 (it starts at 47:07,…
violeta
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Surface of revolution with zero mean curvature

I want to show that the only surfaces of revolution with zero mean curvature are the plane and the catenoid (revolving $x = \frac{1}{a}\cosh(ay+b)$ about $y$-axis). Aiming to do this, I have calculated the first and the second fundamental form for…
Kelvin Hsu
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