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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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Particular function in proof of flow box theorem

Flow Box Theorem If $M$ is a manifold of dimension $n$ and $X$ is a vector field on $M$ such that for a certain $p\in M$ $X(p)\neq0$, then there exists a chart $(U,\phi)$ on $M$ such that $p\in U$ and $\phi_\ast X$, the pushforward of $X$ by $\phi$…
MickG
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Find $\nabla_{\gamma'(t)}\gamma'(t)$. A metric on $\mathbb{R}^2$ is given by the form $dr^2+ f(r,\theta)d\theta ^{2}$ in polar coordinates.

A metric on $\mathbb{R}^2$ is given by the form $dr^2+ f(r,\theta)d\theta ^{2}$ in polar coordinates. Let $\gamma(t)$ be a curve in $\mathbb{R}^2$ given by $\gamma(t) = (t,\theta_0)$ in polar coordinates where $\theta_0$ is a constant. Find…
1233dfv
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Zero Gauss curvature and constant mean curvature of a ruled surface in $\mathbb{R^3}$ implies it is a right cylinder

Assuming I have a ruled surface parametrized as $x(u,v)=\beta(u)+v\delta(u)$, with zero Gauss curvature, which in this case is given by $K=-\frac{m^2}{EG-F^2}$=$\frac{- (\beta'\delta \times \delta')^2 }{|\beta' \times \delta +v\delta\times…
Ali
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What does a skew second fundamental form geometrically mean?

Could there be a realizable 2-surface in some higher dimensional non-Riemannian embedding space whose second fundamental form is skew? If yes, then what would its skew part mean?
Ayan
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Constant torsion-expression of unit speed curves

I am currently studying for an exam in differential geometry. There's a problem which I am not able to solve and do not even know where to start (although I think it has to do with the Frenet equations) :Let $\alpha(s)$ be a curve parametrised by…
mich95
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Connectedness of level sets

I have a $C^{1}$ real valued function $f$ defined on a connected manifold $M$, it doesn't have critical points, lets assume that $f^{-1}(0)$ is a (compact) connected submanifold of $M$, does that imply that every level set will be connected?
Miguel
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differentiable structure on mobius strip

Define $M= \mathbb{R}^2/\sim$ where $(x,y)\sim(x',y')$ if $x-x'=2n$ for some integer $n$ and $y = (-1)^n y'$. Then how can I give a differentiable sturucture on $M$? Is there a general technique for this? (Currently I'm reading Lee's SM. I hope you…
le4m
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a tangent vector which belongs to intersection of a manifold and a subspace is tangent to their intersection?

I have the following situation: There is an embedded submanifold (in my particular case it is of codimension one but I do not think it matters here) $M\subseteq R^n$, and a vector subspace $P\subsetneq R^n$. There are also a point $p\in P\cap M $…
Asaf Shachar
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Is it possible to construct a non-closed plane curve from a closed space curve via the curvature of the latter one

I'm really stuck on this problem. Let $\alpha:[a,b]\subset \mathbb{R}\to \mathbb{R}^3$ be a smooth arc-length parametrized curve and let $\kappa:[a,b]\to \mathbb{R}$ be its curvature. I know from the "Fundamental theorem of the local theory of…
DonQuixote
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Calculus on Manifolds/Differential Geometry

I'm trying to understand differential forms. My instructor explained them in terms of operators on vector fields that spit out continuous functions. I was trying to understand the geometric meaning of them but he says in his notes "take a…
Crunch
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Smooth curves on manifolds

Assume that there is a parametrized smooth curve $c$ on the manifold $M$, mapping from $[a,b]$ to $M$. Also assume that there is a tangent vector on $M$ in the form $(p,v)$. Tu's text states that it is assumed that the curve $c$ is starting at $p$…
Marion Crane
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Hodge star related question

If we have expression (1) $$\star (F \wedge d\alpha)$$ where $$ (F \wedge d\alpha)$$ is a $2$-form field strength ($F$ and $d\alpha$ are 1 forms)and $\star$ represents Hodge star. How can we simplify this rather more? I want to exterior derive…
PhilosophicalPhysics
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Pre-requisites for studying differential geometry?

I am an 3rd year undergrad interested in mathematics.i had read h.graham flengg (from geometry to topology).i found this field interesting i now want to read further,so i want to ask these questions i) prerequisites to read differential…
Nebo Alex
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Flows of two vector fields

Suppose $M$ is a manifold and $f:M \rightarrow \bf{R}$ is a $\mathcal{C}^{\infty}$ function. Let $X$ and $Y$ denote vector fields on $M$, and let $\varphi_Y^t$ denote the flow of $Y$. Fix a point $p \in M$, and consider the real valued function…
Jim K
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Integrating the Riemannian volume form

Let $M$ be a compact manifold with $\partial M = \varnothing$ and let $\omega$ be the volume form $\sqrt{\det g_{ij}} dx_1 \wedge \dots \wedge dx_n$. I want to show that $\omega$ is not exact. My thoughts so far: Assume that there was an $n-1$-form…
self-learner
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