Mathematics Stack Exchange
Mathematics Stack Exchange

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
2
votes
1 answer

Find an ellipse whose length is the same as the outer rim of the monkey saddle

Given the monkey saddle $z=x^3-3xy^2$ over the unit circle $x^2+y^2 \leq 1$, find an ellipse whose length is the same as the length of the outer edge of the monkey saddle. I've already found a parameterization for the monkey saddle in…
zzzzzzzzzzz
  • 1,072
2
votes
1 answer

Non-smooth internal function of composition which happens to be smooth

This is a problem which addresses the dilemma of what can happen of we restrict the conditions for the definition of a smooth function between manifolds. Firstly, the definition of a smooth function between manifolds is that if we have a function…
D.S.
  • 115
2
votes
0 answers

Cartan's Structure Equations

I've encountered the problem which I believe intends to address Cartan structure equations, while reading the book Modern Geometry by Dubrovin, Fomenko and Novikov. It goes like: (Here the summation is understood to take place over repeated…
Kelvin Hsu
  • 137
2
votes
1 answer

Gauss normal map

If $M^{2} \subset \mathbb{R}^{3}$ is a surface with given normal field, we define the Gauss (normal) map $$n:M^{2} \rightarrow \text{unit sphere}\ S^{2}$$ by $$n(p) = \textbf{N}(p), \qquad \text{the unit normal to $M$ at $p$}.$$ Why use a unit…
nightmarish
  • 617
2
votes
1 answer

Every manifold is locally the inverse image of a regular value

Suppose $M \subset \mathbb{R}^{m+d}$ is a manifold of dimension $m$. I need to prove that $M$ is locally $g^{-1}(0)$ where $g:U \subset M \rightarrow \mathbb{R}^p$ is a submersion. I am suppose to use the fact that every manifold is locally…
allizdog
  • 897
2
votes
1 answer

Parametrization of a Parabola's Evolute

Let $y=x^2/2$. Its parametric form is $r(t)=t\,\hat i+t^2/2\,\hat j$, and its evolute is $$ c(t)=-t^3\,\hat i+\frac{3t^2+2}{2}\,\hat j.\tag{1} $$ Visually,                                          When I rewrite $(1)$ as a normal function, by…
wjmolina
  • 6,218
  • 5
  • 45
  • 96
2
votes
0 answers

Using Gauss-Bonnet in a Proof

I was wondering if there was a prove of the Poincaré-Hopf index theorem using Gauss-Bonnet I'll really like to see one to understand it better.
HelaHelheim
  • 169
2
votes
1 answer

Is there a canonical connection on an abstract smooth connected manifold?

Q1) Something about the idea of a connection on the tangent bundle of a manifold confuses me. Naïvely, given an abstract smooth connected manifold $M$ of dimension $m$ we want to "connect" the tangent spaces so that we can define, say, a covariant…
J. Dong
  • 47
2
votes
1 answer

The integral curves of the vector field given by the gradient of $f(x,y,z)=z$

Suppose you have a function $f: S^2 \rightarrow \mathbb{R}$ given by $f(x,y,z)=z$. Then the gradient of $f$ defines a vector field over the sphere. My question is what are the integral curves of this vector field? I have tried this: If $c: J…
allizdog
  • 897
2
votes
0 answers

Determine if a surface exists with given First and Second Fundamental Form coefficients

I know I can use Theorema Egregium to determine if a surface DNE. However using this method, I can convince myself that I can prove exisitence. For example, I'm now working with a question given $E=1$, $F=0$, $G=\cos^2u$ in First Fundamental Form…
MonkeyKing
  • 3,178
2
votes
1 answer

Do Killing vectors satisfy $\nabla_a\nabla_bK_c + \nabla_b\nabla_cK_a + \nabla_c\nabla_aK_b = 0$?

Suppose $K$ is a Killing vector satisfying Killing's equation, $\nabla_bK_a+\nabla_aK_b=0$. As part of a larger problem, I am wondering if I have correctly shown that $$\nabla_a\nabla_bK_c + \nabla_b\nabla_cK_a + \nabla_c\nabla_aK_b = 0$$ My…
Doubt
  • 1,719
2
votes
1 answer

Show that the map is a diffeomorphism

Given a set $M = \{(x,y,z)\in \mathbb R^3 | x^2+y^2-z^2=1\}$ and a function $\alpha \colon M \to S^1 \times \mathbb R$, $$\alpha(x, y, z) = \left(\frac{(x,y)}{\sqrt{1+z^2}},z\right)$$ Is there a smart method of showing that $\alpha$ is a…
guest
  • 645
2
votes
1 answer

Show that a surface is regular

The helicoid is the surface swept out by a rotating horizontal line as it rises along the $z$-axis (see the figure below). It can be described by the parameterized surface $x : \mathbb{R}^2 \to \mathbb{R}^3$, $x(u, v) = (av\cos u, av\sin u, bu)$,…
Elay Marco
2
votes
0 answers

Tangent space at function of torus and circle

Consider the torus $S^1 \times S^1 := [(x_1,x_2,x_3,x_4) \in E^4 \space | \space x_1^2+x_2^2=1,\space x_3^2+x_4^2=1]$. And the sphere $S^2 := [(x_1,x_2,x_3) \in E^4 \space | \space x_1^2+x_2^2+x_3^2=1]$. Let $f: S^1 \times S^1 \to S^2$ be the…
Martin
  • 127
2
votes
1 answer

Question about a lemma concerning vector fields in Do Carmo

In the book Differential Geometry of Curves and Surfaces by Do Carmo, he gives a proof of the following lemma: Let $w$ be a vector field in an open set $U \subset \mathbb{R}^{2}$ and let $p \in U$ be such that $w(p) \neq 0$. Then there exist $W…
user135520
  • 2,137
1 2 3
…
99 100