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
2 answers

Parametrization of one-sheet hyperboloid

I was given the following exercise: let $S$ be $x^2 +y^2-z^2=1$. show that for every real number $t$ the line $l_t$ $$(x-z)\cos t=(1-y)\sin t,\quad (x+z)\sin t=(1+y)\cos t$$ is contained in $S$; show that every point of $S$ is contained in one…
Mockingbird
  • 983
2
votes
1 answer

Intersection of surface and tangent plane around hyperbolic point

"Show that if $P$ is a hyperbolic point, a neighborhood of $P$ in $M \cap T_P M$ is a curve that crosses itself at $P$ and whose tangent directions at $P$ are the asymptotic directions." This is question 2.2.21c of Shifrin's Differential Geometry…
philippe
  • 33
2
votes
0 answers

Help me complete this Differential Geometry equation for a geodesic.

I am wondering what happens to the geodesic equation if there is a constant acceleration,$A_3$: $$\frac{d^2x^\nu}{ds^2}=A_3$$ $$\frac{dx^\nu}{ds}=\frac{\partial x^\nu}{\partial x^\beta}\frac{d…
Quark Soup
  • 233
2
votes
0 answers

A Problem from Docarmo Differential Geometry chapter 2.4

It's number 23 in chapter 2-4(The tangent plane). The Problem is, Let $P: C \rightarrow C$ be the complex polynomial $P(\phi)=a_0\phi^n +a_1\phi^{n-1}+ \cdots +a_n $, $a_0 \not= 0$, $a_i \in {C}$ Denote by $\pi_N$ the stereographic projection of…
user773114
  • 21
2
votes
0 answers

quaternionic contact structure

I've been reading about quaternionic contact structures, but struggling to understand how they can be thought of as a G-structure (in this case $G=Sp(n)Sp(1)$). For context, the definition I'm taking for a quaternionic contact structure is a…
user345730091
  • 279
2
votes
1 answer

Unique Perpendicular Geodesic

Let $p < q < r < s$ be real numbers. Let $l$ be the geodesic with endpoints at $p$ and $q$ and let $m$ be the geodesic with endpoints at $r$ and $s$. (a) Prove that there is a unique geodesic segment from $l$ to $m$ that is perpendicular to both.…
Timothy Bade
  • 21
2
votes
0 answers

characterization of the differentiable functions over a regular surface

Let $S$ be a regular surface. And let $f:S\to \mathbb R$ be a differentiable function It's not hard to prove that if $ W$ is an open set of $\mathbb R^3$ such $ V\subset S\subset W$, and $f:W\mathbb \to R$ is a differentiable function (in the…
Miguel
  • 459
2
votes
1 answer

What is the geometric reason for the negative sign in front of the scalar product defining the second fundamental form?

This is bound to have some easy explanation in terms of aligning the differential of the normal vector to the surface $S$ with the direction of the vector taken into the quadratic $\vec v$, but I am not sure why. Since it is a definition, the…
Antoni Parellada
  • 8,889
2
votes
1 answer

Parallelizability of 2-manifolds

I have found this cute problem: Let's $M$ a 2-dimensional orientable manifold which has a non zero vector field $X$. Show that $M$ is always parallelizable. My idea was consider $p\in M$ so $X(p)\in T_pM$ is a non zero vector. Then I can complete…
Protagora1789
  • 31
2
votes
1 answer

What’s is the 1st Fundamental form? The coefficients or the dot product?

I often hear that the coefficients of the 1st FF are the 1st FF. But the I also hear that the dot product is the 1st FF. But make sense to me but...which one is which? If the dot product is the 1st FF, then how come people say that Gaussian…
Spencer Kraisler
  • 2,882
2
votes
0 answers

Torsion of a closed curve

Why the total torsion of a closed curve on a sphere is zero? What is the meaning of total torsion? Is it mean that we should calculate the torsion from point A to point A (i.e. since the curve is closed, we calculate the torsion from the start…
Lila
  • 73
2
votes
0 answers

Does the derivative of the derivative depend on a choice of connection?

Let $X,Y$ be smooth manifolds and consider the infinite-dimensional manifold $$ C^\infty(X,Y) $$ of smooths maps $f: X \to Y$. Note that there is an infinite-dimensional vector bundle $E$ over this space whose fibers are given by $f^\ast(TY)$-valued…
user54535
2
votes
0 answers

showing that any curve is a geodesic on the surface generated by its binormals

Show that any curve is a geodesic on the surface generated by its binormals. Normal property says that, any curve $u=u(s),v=v(s)$ on a surface $\textbf{r}=\textbf{r}(u,v)$ is a geodesic iff the principal normal at every point on the curve is…
am_11235...
  • 2,142
2
votes
3 answers

Given two points in a manifold, can one always find a topological ball that contains both?

I hope you will bear with me and excuse me if my question is kind of obvious to many of you. In an $n$-manifold say we have $x,y$ in it. Could we always find an open $n$ dimensional ball contained in the manifold such as both points belong to it?
El Moro
  • 1,571
2
votes
1 answer

Easy way to determine the sign of Gaussian curvature without explicit computation.

I know how to compute Gaussian curvature via the first and second fundamental form. i.e., \begin{align} K = \frac{eg-f^2}{EG-F^2} \end{align} In this computation one usually set $n = \frac{X_u\times X_v}{||X_u\times X_v||}$ Without detail…
phy_math
  • 6,448
1 2 3
…
99 100