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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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parametrization of the hyperboloid of two sheets

Find the parametrization for the hyperboloid of two sheets${(x,y,z) \in \mathbb{R}^3}; -x^2-y^2+z^2=1$. Ok so I saw two answers for this question: $x(u,v)=(\sinh u \cos v, \sinh u \sin v, \cosh u)$ and $x=(u,v)= (\cosh u \sinh v, \sinh v, \cosh u…
Ruth Gutierrez
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Tangent bundle of the projective space.

I would like to know how does one imagine/write-down the tangent bundle of the real/complex projective space. Is there something simplifying that happens especially for $\mathbb{RP}^1$? Isn't there some relationship of this tangent bundle to…
Student
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Given the curvature and torsion, find the curve

I need some help on the following problem: Given that a curve $\mathbf r:I\to \Bbb R ^3$ has constant curvature $k(s)=k$, for all $s$, and constant torsion $\tau(s)=\tau$, for all $s$. Find the curve $\mathbf r$. I only know that, according to…
Scorpio19891119
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Killing vector field of constant length on Riemannian manifolds

I would like to solve next problem A Killing vector field $X$ on a Riemannian manifold $(M, g)$ ($g$ is metric) has constant length if and only if every integral curve of the field $X$ is a geodesic in $(M, g)$. I found here…
user110822
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Christoffel Symbols Equality Solution?

They changed the exercise, so I tried to solve it again: I have to prove the following: Let $\Omega \subseteq \mathbb{R}^d$ be open and $g$ a metric field on $\Omega$. For every $\phi \in \mathrm{Diff}(\Omega)$ let $\Xi^i_{jk}[\phi]$ be functions on…
b00n heT
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Hodge double star operator

I want to prove that $**\omega=\left(-1\right)^{k\left(n-k\right)}\omega$, where $*$ is the Hodge star operator acting on differential $k$-forms $\omega$ on $\mathbb{R}^n$. Where can I find the proof of this?
Alem
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General geodesics

How to solve the following: Let $f : (M,\nabla)\rightarrow (\overline{M},\overline{\nabla})$ be a diffeomorphism of manifolds with torsion-free connections. a) For reparametrisation $\alpha$ of geodesic line on M holds…
alans
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Dupin's indicatrix of the monkey saddle

The "monkey saddle" is a parametric surface defined by $$ \begin{eqnarray} x & = & u \\ y & = & v \\ z & = & u^3 - 3 v^2 u \end{eqnarray} $$ Its second fundamental form has the coefficients $e = f = g = 0$ at the origin $(0, 0, 0$). The Dupin…
koletenbert
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plane of symmetrie leads to geodesic

Any plane of symmetry intersects a surface in a geodesic. I am having a little difficulty with the proof of this one. The proof says: The normal to the surface in such a point must be invariant under reflection in the plane of symmetry and hence…
dinosaur
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Flat Lie algebras

I would like to review a classical result by Milnor, Curvatures of left invariant metrics on Lie groups. J., Adv. Math. 21 (1976), no. 3, 293-329. Theorem 1.5 (page 298). A Lie group with left invariant metric is flat if and only if the associated…
amine
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What is the modulus of a tensor on a Riemannian 3-manifold?

Let $v^i$ be a vector on a Riemannian 3-manifold with metric $g_{ij}$ embedded inside a 3+1 space-time such that for some constant $N_M$ it satisfies the inequality $g_{ij}v^iv^j \leq N_M ^2$. Let $K$ be a symmetric rank-2 tensor on the 3-manifold.…
Student
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Does $L^2$ commute with Hom?

Let $E,F \to M$ be two smooth vector bundles over a compact manifold $M$. It is well-known that the homomorphism fields $Hom(E, F) \to M$ are a smooth vector bundle, too. In fact, this bundle can be thought of as $E^* \otimes F$. Denote by $\Gamma(…
Meneldur
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parametrization and orientation of the Möbius band

I know that the Möbius band is a nonorientable surface. However, the following exercise seems to contradict this. A Möbius band can be constructed as a ruled surface by $x(u,v)=\beta(u)+v\gamma(u)$, where $-1/3
noot
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Fundamental theorem of calculus and pullback bundles

Let $X$ be a manifold and $\pi:E\rightarrow X$ a vector bundle over $X$ equipped with a metric $\left\langle \cdot,\cdot\right\rangle $. Let $f:[0,1]\rightarrow M$ be a smooth map, and consider the pullback bundle $f^{*}E\rightarrow[0,1]$. This is…
fran
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Vector calculus and Frenet-Serret equations

I have shown the first two equality and I am working on the showing the 1st equals the 3rd. \begin{alignat*}{4} \frac{1}{\rho}\hat{\mathbf{{n}}} &= \frac{d\hat{\mathbf{{u}}}}{ds} &{}= \frac{\dot{\hat{\mathbf{{u}}}}}{\dot{s}} &{}=…
dustin
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