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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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Find a parametrization of the intersection curve between two surfaces in $\mathbb{R^3}$ $x^2+y^2+z^2=1$ and $x^2+y^2=x$.

Find a parametrization of the intersection curve between two surfaces in $\mathbb{R}^3$ $$x^2+y^2+z^2=1$$ and $$x^2+y^2=x.$$ I know that $x^2+y^2+z^2=1$ is a sphere and that $x^2+y^2=x$ is a circular cylinder. Any help is greatly appreciated, thank…
Gino
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Exterior product of a differential form and its derivative

Let $\omega$ be a k-form on a smooth manifold $M$ such that there exists $f\in C^{\infty}(M)$ with $f(x)\ne 0$ for all $x\in M$ and $d(f \cdot \omega)=0$. I need to show that $\omega \wedge d\omega =0$. I have only been able to show that $\omega…
Nitrogen
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Special types of Sasaki manifolds

i have a question to special cases of Sasaki-manifolds. Let $(M, g, \xi, \eta, \Phi)$ a Sasaki-manifold. In special case maybe $M=S^{2m+1} \cong \mathbb{C}^{m+1}$. This is a Sasaki manifold. But what is $\Phi$? Analogous: Let $M=\mathbb{R}^{2m+1}$,…
Ronald Müller
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global function defining a $\mathcal{C}^1$ submanifold

Let $M$ be a $\mathcal{C}^1$ submanifold of dimension $1$ of $\mathbb{R}^2$. Then for each $x\in M$, there is a neighbourhood $U$ of $x$ and a $\mathcal{C}^1$ function $f:U\to \mathbb{R}$ such that $M\cap U=f^{-1}(\{0\})\cap U$ and $\nabla f\neq 0$…
kaleidoscop
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Space Curve: Equation of normal plane at point...

If I have the space curve $r(t) = \langle t, t^2, t^3 \rangle$, how would I find an equation of the normal plane to $r(t)$ at the point $P(2,4,8)$?
penu
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How to define a vector field passing through a given point.

Given a manifold $M$ we can consider its tangent bundle $TM$. Fix $m \in M$ and $v \in TM$. Is it always possible to define a vector field $F$ such that $F(p)=v$?
sss
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Prove that orientable surface has differentiable normal vector

Prove that: a regular surface $S\subset \mathbb{R}^3$ is an orientable manifold if and only if there exists a differentiable mapping of $N:S\rightarrow \mathbb{R}^3$ with $N(p)\perp T_p(S)$ and $|N(p)|=1$, for all $p\in S$. If part: I…
JSCB
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Gaussian Curvature of Pseudosphere

I have the parametrization $x(u,v)=(\cos u \sin v, \sin u \sin v , \cos v+\log (\tan {v/2}))$ with $0
EQJ
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Surjectivity of the pullback

I've tried to prove the naturality of the pullback of a connection. I've reduced it to the following question: Is the pullback a surjective mapping on the space of sections of a vector bundle? i.e., suppose I have a smooth vector bundle $~E\to M$,…
David Roberts
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How to obtain the standard basis for $\mathbb{R}^2$ using differential geometry?

I apologize in advance because I don't know how to enter code to format equations, and I apologize for how elementary this question is. I am trying to teach myself some differential geometry, and it is helpful to apply it to a simple case, but that…
Sam
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The properness of a submersion

Let $M$ and $N$ be two differential manifolds and there is a surjective submersion $f$ from $M$ to $N$ such that $f^{-1}(p)$ is compact and connected for any $p$ on $N$. Can we conclude that $f$ is proper, that is, the preimage of a compact set is…
Summer
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Exterior derivative of local basis element $dx^k$ is zero

Let $M$ be a smooth manifold and let $\omega = \sum_{(i_1, \dots, i_n)}f_{(i_1, \dots, i_p)} dx^{i_1} \wedge \dots \wedge dx^{i_p}$ be a differential $p$-form. Let $d$ denote the exterior derivative. I am trying to show that $d(dx^k) = 0$. To this…
self-learner
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Compare three tangent vectors constructed via parallel transport, exponential map and Jacobi vector field respectively

Given a Riemannian manifold $X$, a point $x\in X$ and $u,v\in T_xX$, I wanted to compare the following three vectors in $T_{\exp_x(v)}X$. $u_1=$ The parallel transport of $u$ along the geodesic $\exp_x(tv)$ at $t=1$; $u_2=(d\exp_x)|_{\exp_x(v)}u$,…
Xin Jin
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Getting Ricci Curvature From $g_{ab,cd}$

How does one see that $$c g_{ab,cd}\left(\eta^{ac}\eta^{bd} - \eta^{ab}\eta^{cd}\right)$$ is equal to $$(c/2)\eta^{bc}\eta^{ae}\partial_{a}\left(g_{be,c} + g_{ce,b} - g_{bc,e}\right) - (c/2)\eta^{ae}\eta^{bc}\partial_{c}\left(g_{be,a} + g_{ae,b} -…
bobby
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Gaussian curvature of one sheet hyperboloid

Q: Consider an one sheet hyperboloid $S$ sitting in $\mathbb{R}^3$ which defined by $x^2+y^2-z^2 =1$. Show that there is a straight line in $S$ through every point of $S$. Also, deduce without any calculation, that the Gaussian curvature of $S$ is…
SamC
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