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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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Definition of Willmore energy

The MAA has posted to its facebook page a link to an article about a recent proposed proof of what is called the Willmore conjecture, after Thomas Willmore. Wikipedia's article titled Wilmore conjecture includes the following: Let…
Michael Hardy
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The extension of diffeomorphism

Let ${\Omega _1}$,${\Omega _2}$ be two open sets in $\mathbb R^n$ and $f$ is a diffeomorphism between them. For every $x$ in ${\Omega _1}$, is there an open set $\Omega_{x} \subset \Omega_1$ and a diffeomorphism $g$ of $\mathbb R^n$ such that $g=f$…
Summer
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Relationship between Curvature and Torsion on a Sphere [Homework]

The problem asks: Let $\kappa(s)$ and $\tau(s)$ be the curvature and torsion of a curve parameterized by its arclength $s$. If $$(\frac{1}{\kappa})^2 + [\frac{1}{\tau}\frac{d}{ds}(\frac{1}{\kappa})]^2 = constant$$ , then either this curve lies on a…
taper
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How do connection 1-form and Ehresmann version of connections relate to each other?

I am studying connections on abstract manifolds. So far, I have read several equivalent definitions but I can't establish the equivalence between them on my own. The first definition is the Ehresmann connection that defines a connection on a…
user246836
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Is it meaningful to take "exterior products" of vector fields?

Let $M$ denote a smooth manifold. I've read that a differential $k$-form is a smooth section of the $k$th exterior power of the cotangent bundle of $M$. However I barely understand what this means, and I'm trying to understand it better by tinkering…
goblin GONE
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First variation of an action?

I'm working on a problem and I must compute the first variation of an action. Let $\Omega$ is a 2-form on a semi-Riemannian manifold $M$ and $f$ is a smooth function and $\Gamma$ is an 1-form on $M$. I obtained the following…
ramandi
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Stokes theorem for Lorentz manifolds

Reading Tao's book: Nonlinear Dispersive Equations I came upon an identity (the energy flux identity for the wave equation, page 90) for which the proof uses the Stokes theorem. In this case he uses the Stokes theorem on a truncated backward…
i like xkcd
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Closed and Exact forms/deRham groups

I'm trying to translate these theorems, below, into theorems about vector and scalar fields in $\mathbb R^n\setminus\{0\}$, in the case $n = 2$. First Theorem: Let $A = \mathbb R^n\setminus \{0\}$, with $n \ge 1$ a) If $k$ is not equal to $n-1$ then…
mary
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Making a bijection into a diffeomorphism

Given a set $M$, one that can be made into a smooth manifold, and a bijection $f:M\to M$, does there exist a differentiable structure on $M$ such that $f$ is a diffeomorphism? In case it's not always true, what should $M$ and $f$ satisfy in order to…
user251131
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Interior Derivative and Contraction: Kobayashi and Nomizu.

In Kobayashi & Nomizu, the interior derivative of an r-form is defined as $\iota_X \omega = C(X \otimes \omega)$, where $C$ is the contraction associated with the pair $(1,1)$ and $\omega$ is interpreted as a tensor of type $(0,r)$. They then claim…
TJ2
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Is parallel transport injective?

For a vector bundle $E\to X$ with a given connection $\nabla$. We say that a section $s$ of $E$ is parallel to a vector space $V$ if $\nabla_V s=0$. If $\gamma:[0,1]\to X$ is a smooth path, we say that $s$ is a parallel transport of $v$ along…
Mauricio Tec
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Shouldn't these two definitions for curvature agree?

In $\mathbb R^n$ the defintion of curvature of a smooth regular curve $\gamma : \mathbb R \to \mathbb R^n$ is $$ \kappa (t) = \|\gamma''(t)\| / \|\gamma '(t)\|$$ In $\mathbb R^2$ the definition for the curvature of an arbitrary smooth regular curve…
student
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characterization of non compact surfaces in $\mathbb{R^3}$

Is there a way to characterize non compact surfaces with constant mean and gaussian curvature. I know that if $K=0=H$ then the surface is a plane. How can I know about the others? Just to add, for compact surfaces with constant positive curvature I…
Ali
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Is it true that a map that maps geodesics to geodesics must be an differeomorphism?

Let $M,N$ be connected Riemannian manifolds. Let $f:M\rightarrow N$ be a bijective smooth map that maps any unit speed geodesic in $M$ to unit speed geodesic in $N$. Question: Is this suffice to show that $f$ induces a differeomorphism $M\cong N$?…
Bombyx mori
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Left-Invariant Vector Fields: Smoothness

Given a Lie group $G$. For a left-invariant vector field it holds: $$\mathrm{d}l_gV=V\circ l_g:\quad V_g=\mathrm{d}l_gV_e$$ Conversely rough vector fields are smooth: $$V_g:=\mathrm{d}l_gv:\quad V\in\Gamma_G(\mathrm{T}G)$$ How to prove this in a…
C-star-W-star
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