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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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gauss map takes geodesics to geodesics

Let $S$ be a regular surface, and let's consider $\gamma: I \to S$ be a geodesic. Let $ N: S \to S^2 $ be the gauss map. Then $ \beta(s) = N(\alpha(s))$ is a curve $\beta : I \to S^2$ (where $S^2$ denotes the unit sphere). I want to prove that…
Trafalgar Law
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second fundamental form of a surface given by regular values

Consider the differentiable function $f:\mathbb R^3 \to \mathbb R$. Consider the regular surface given by $ f(x,y,z)=a$ where $a$ is a regular value of $f$. Prove that the second fundamental form of the surface is given by: $$ \text{II} \left( v…
Trafalgar Law
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exact differential n-forms

We know that a 1-form $\omega$ on a manifold $M$ is exact if and only if $\int_{\gamma}\omega=0$ for any closed loop $\gamma$. How can I prove the following generalization: $\omega$ is an exact n-form on $S^n$ if and only if $\int_{S^n}\omega=0$?…
Manuel
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Is $xyz=k^3$ a developable surface

I don't know how to judge whether it's a developable surface or not.I think firstly I should find a way to write the surface as a form of a ruled surface,that is,write the surface something like $$r(u,v)=a(u)+vl(u)$$,but setting $x=u,y=v,z=k^3/uv$ I…
math
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Many questions about the Chern-Weil theory and the proof of the Bott vanishing theorem

I'm reading 'Lectures on Chern-Weil theory and Witten deformations' by Weiping Zhang. Now I have many questions: He defined $tr:\Omega^*(M,End(E))\to\Omega^*(M)$ by $\omega A\mapsto\omega tr[A]$ where $tr[A]$ as fiberwise to be a smooth function.…
WakeUp-X.Liu
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When a smooth curve is an immersion (John Lee's Smooth manifold book p 156) and Example 7.3

In John Lee's Intro to Smooth Manifold book (2003 Springer) , I need some help with an example of an immersion. On page 156 Example 7.1 c), If $\gamma(t): J \to M$ is a smooth curve ...then $\gamma$ is an immersion if and only if $\gamma'(t)\neq 0 $…
xbl
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About the Gauss Equation and the Codazzi Equation

I am recently taking the undergraduate version of differential geometry. This week we finished the Compatibility Equations. Actually the contents given in the lecture are quite different from the book I read, Differential Geometry of Curves and…
Siamese
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Is this a bundle?

In Frederic Schuller's lecture series Lectures on The Geometrical Anatomy of Theoretical Physics, he gives an example of a bundle $E\overset{\pi}{\rightarrow}M$ where different points of the base manifold have different fibres: Or, as spelled out…
Davide Radaelli
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Local Reparametrization of Surface using known Vector Field (Differential Geometry)

I need help with the following problem: "Let $X$ be a vector field defined on surface $S$, and $p \in S$ such that $X(p) \neq 0$. Prove that there exists a local parametrization $\phi \colon U \to S$ with $U$ an open set of $\mathbb{R}^2$ such that…
Jarana
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Angle between two lines on parametric surface

I need to find what angle is between the lines $u=-v$ and $u=v$ on the surface: $$r(u,v)=(u\cos(v), u\sin(v), 2v)$$ The only thing that comes to my mind is to put both line equations to the surface equation: $$…
Andrew
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Quick question: Poincare dual class to the fundamental cycle of a submanifold

$X$ is a compact, oriented $n$-manifold and $Y$ is a $k$-submanifold of $X$. Let $\eta_{Y}$ be the Poincare dual class to the fundamental cycle $[Y] \in H_{k}(X,\mathbb{Z})$ and $\phi \in H^{k}_{dR}(X)$. Why is $\int_{Y} \phi = \int_{X} \eta_{Y}…
lnth
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Computing integral of $2$ - form on a torus

I am looking at problem 16-2 of Lee's Smooth Manifolds, second edition. Problem 16 - 2: Let $\Bbb{T}^2 \subseteq \Bbb{R}^4$ be the two torus defined as the set of points $(w,x,y,z)$ such that $w^2 + x^2 = y^2 + z^2 = 1$, with the product…
user38268
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How to define a partial derivative invariantly?

Let $M$ be a smooth manifold and $f$, $g$ be smooth functions in some neibourhood of a point $x_0\in M$, $\nabla g\ne0$. 1) How to define $\displaystyle \frac{\partial f}{\partial g}$ invariantly? If $M$ is a domaqin in $\mathbb R^n$ then the…
Andrew
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$y^2 - x^3$ not an embedded submanifold

How can I show that the cuspidal cubic $y^2 = x^3$ is not an embedded submanifold of $\Bbb{R}^2$? By embedded submanifold I mean a topological manifold in the subspace topology equipped with a smooth structure such that the inclusion of the curve…
user23086
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Intuition regarding Euler’s formula

In every beginner’s class on differential geometry, we learn Euler’s formula, which tells us about the normal curvatures at a point on a surface. I know how to prove this formula, and I have even taught it in classes, but it seems entirely…
bubba
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