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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).

32835 questions
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Show that F(t) is an immersion

I've got here an exercise that says: "Show that the map $F:\mathbb{R}\rightarrow \mathbb{R^2}$ defined by $F(t)=(\cos t, \sin t)$ is an immersion". $F$ is an immersion if $dF_x:T_x\mathbb{R}\rightarrow T_{F(x)}\mathbb{R^2}$ is injective. Now $dF_x$…
batman
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Quotient map action on commuting vector fields

Let $T^m = \mathbb{R}^m / \mathbb{Z}^m$ be the $m$-dimensional torus equipped with quotient topology, so the canonical surjection $\pi: \mathbb{R}^m \to T^m$ is a covering map. Show that there is an atlas on $T^m$ such that $\pi$ is a smooth…
Eric Auld
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De Rham Cohomology of Hopf Surface

How I calculate the De Rham coohomology of the Hopf surface? In particular I would like to know why the second Betti number is zero .
user52342
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Covariant derivative of (1,1)-tensor

Suppose I have an endomorphism $J:TM \to TM$ and a connection on M. It is possible to define $\nabla_X J$ by transforming $J$ into a (1,1)-tensor and using the extension of $\nabla$ to tensors. Going back we get an endomorphism $\nabla_X J:TM \to…
Lucas Kaufmann
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Covariant derivative with contravariant components derivation

I'm doing Leonard Susskind's course on General Relativity (http://deimos3.apple.com/WebObjects/Core.woa/Feed/itunes.stanford.edu-dz.19344853322.019344853324 ), and I'm stuck on a particular derivation (in Lecture 4, about 12 minutes in). He left as…
Extropy
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The non-vanishing 1-form on $\mathbb R^2$

If $\omega$ is a non-vanishing 1-form on $\mathbb R^2$, then for any a point $p\in \mathbb R^2$, can we find an open neighborhood $U$ of $p$ and two functions $f,g$ on $U$ such that $\omega=fdg$ on $U$?
Summer
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Gauss map and shape operator

Define the map $$\pi : (\mathbb{R}^3-\{(0,0,0)\})\to S^2$$ by $\pi(p)=\frac{p}{||p||}.$ Show that if $\Sigma_R$ is the sphere of radius $R>0$, then the Gauss map of $\Sigma_R$ is $\pi|_{\Sigma_R}$ (which means the map $\pi$ restricted to the…
Lays
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Change of coordinates with Jacobian

We know that the change of variable in $\mathbb{R^n}$ with a $T: V \to U$ is a diffeomorphism of open sets in $\mathbb{R^n}$ and $f$ is an integrable function on $U$. Then $$\int_U f dx_1 \cdots dx_k = \int_V (f \circ T) |\det(dT)|dy_1 \cdots…
Rosa Maria Gtz.
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Is a cylinder a Lipschitz domain?

I'm wondering if the domain $(0,T)\times \Omega$ is a Lipschitz domain ($T$ is a positive real number), provided that $\Omega$ is an open bounded subset of $\mathbb{R}^n$ with Lipschitz boundary, and how to prove or disprove this fact. Thank you
user37238
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Arc length parameterization lying on a sphere

Show that if $\alpha$ is an arc length parameterization of a curve $C$ which lies on a sphere of radius $R$ about the origin then $$R^2 = (\frac{1}{\kappa(s)})^2+((\frac{1}{\kappa(s)})'\frac{1}{\tau(s)})^2.$$ I know I can write the unit tangent…
Lays
  • 2,023
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Is an Exterior Product the "Opposite" of an Inner Product

One is represented by a dot product, the other by a cross product. The "inner product collapses two co-ordinate vectors into a scalar, the exterior product seems to expand them in a multilinear (manifold)? The inner product seldom has…
Tom Au
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Calculate the curvature of a parametrized curve

I have started to study differential geometry and have some questions about an exercise which is probably not very difficult. Exercise: Let $\gamma: I \rightarrow\mathbb{R}^{2}$ be a regular curve, parametrized by arclength, with Frenet frame $\{…
El_Loco
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Values of the Christoffel symbols

Are the values of the christoffel symbols the same for all coordinate systems on a surface/manifold? I would love to see an example for the cone in two different parametrizations.
BinaryBurst
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Why does my tangent vector not lie in the tangent space?

Me again, still learning my lesson of "don't drink and derive": I have got two parametrizations of the surface $H :=\{ (x,y,z) \in \mathbb{R}^3 \, | \, z^2 = 1+x^2+y^2, \, z > 0\}$, $$F:\mathbb{R}^2 \rightarrow H, \ (x,y) \mapsto…
Embarassed Guy
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Covariant derivative/connection of a local frame $\nabla \bar e = \nabla(e)g + edg$

Suppose $e$ and $\bar e$ are two frames for a vector bundle $E \to M$ over $U \subset M$ such that $\bar e = eg$ for some $g: U \to \text{GL}(r,\Bbb R)$. Then $$\nabla \bar e = \nabla(e)g + edg.$$ I'm trying to understand this which is seemingly…
Tepes
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