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
3
votes
1 answer

$M_m$ is naturally isomorphic to $(F_m/F_m^2)^{*}$

Let us denote $M_m$ be the set of tangent vectors to a manifold $M$ at point $m$ and is called tangent space to $M$ at point $m$ we denote $\bar{F_m}$ be the set of all germs at point $m$ and $F_m$ be the set of germs vanishes at $m$ In warner book…
Myshkin
  • 35,974
  • 27
  • 154
  • 332
3
votes
0 answers

How to show C is a regular curve?

The gradient of a differentiable function $f:S\to \mathbb R$ is a differentiable map grad $f:S\to \mathbb R^3$ which assigns to each point $p\in S$ a vector grad $f(p)\in T_p(S)\subset \mathbb R^3$ such that $$\langle \operatorname{grad}{f(p)},…
user71346
  • 4,171
3
votes
1 answer

Why is L the derivative of L? I have a very vague understanding about the very step needed to show $dL=L$.

I have a bits-and-pieces understanding on how to solve this problem and just a very rough intuition of the path to solve the problem but very much struggling to get to show $dL=L$. This is part of a question if Do Carmo's book that reads as…
user71346
  • 4,171
3
votes
1 answer

Compact Manifold with Geodesic B0undary

Suppose $(M,g)$ is a 2-dimenensional compact Riemannian manifold with boundary $S$. Furthermore, assume that $S$ is totally geodesic. Now consider the conformal change of metric $\widetilde{g}=e^{\phi}g$ for some smooth function $\phi$. In order for…
Andrew
  • 101
3
votes
1 answer

First fundamental form

I am looking at the following exercise of the book of A.Pressley: Let $$Edu^2+2F dudv+Gdv^2$$ be the first fundamental form of a surface patch $\sigma(u, v)$ of a surface $S$. Show that, if $p$ is a point in the image of $\sigma$ and $v, w \in…
user175343
3
votes
0 answers

Ruled surface swept out by normal lines of a curve

The problem is: "Say $\gamma:[a,b] \to \mathbb{R}^3$ is a curve of general type with principal normal vector field $\textbf{n} = t_2$. Show that the ruled surface $r(u,v) = \gamma(u) + v\textbf{n}(u)$ is developable if and only if $\gamma$ is…
Ryan
  • 1,106
3
votes
1 answer

Conformal iff $E=G$ and $F=0$

Prove that a parametrization $x(u,v)$ is conformal (angle-preserving) if and only if the coefficients of the first fundamental form satisfy $E=G$ and $F = 0$. My attempt: It suffices to consider $e_1$ and $e_2$ in $\mathbb{R}^2$. It is easy to…
Nighty
  • 2,152
3
votes
1 answer

Are flat manifolds affine?

I try to understand an article where it is stated that some results regarding affine manifolds apply to the case of the manifold being a flat, compact Lorentzian manifold. The definition of affine, in this context, is that the manifold has a…
Vertex
  • 432
3
votes
0 answers

What's it like researching Differential Geometry

I am an undergraduate and want to go into pure math research once I've finished my degree. I think Differential Geometry looks like a really interesting area and also enjoy the classes I have took in it so far. I am wondering what it is like day to…
Crunch
  • 533
3
votes
1 answer

Projective Space orientation

I'm trying to prove that the projective plane $\mathbb{P}^n$ is orientable is and only if $n$ is odd. To do that that, I have a hint,to prove that the antipodal map is orientation preserving if only if $n$ is odd, I've done that, but it don't know…
Jr.
  • 4,026
3
votes
0 answers

Complete a set of functions to obtain a system.

Let $M^m$ be a $m-$manifold and $C=\{y^1,\dots,y^k:U\subset M\rightarrow \mathbb{R}\},\;k
user31236
3
votes
0 answers

Integral Curve on a Manifold, verification

A curve $\gamma : (a,b)\rightarrow M$ is an integral curve of X if and only if \begin{equation} d\gamma\bigg(\frac{d}{dr}\bigg|_{t}\bigg)= X(\gamma(t)) \end{equation} $t \in (a,b)$ Suppose we require an integral curve through $m\in M$ with…
echelon
  • 165
3
votes
1 answer

Can we define higher-order tangent spaces of a manifold with equivalence classes defined equating second derivatives?

I've been thinking of an obvious way to define higher order tangent space with smooth curves, but it seems that this definition does not coincide with the repeated tangent bundle construction. So, let $M$ be a smooth manifold, and let $\gamma: [0,1]…
sure
  • 849
3
votes
1 answer

Why does the torsion remain invariant under a change of orientation?

There is this statement in the book of Differential Geometry of Curves and Surfaces by Do Carmo page 18 that I have a doubt in. Notice that by changing orientation the binormal vector changes sign, since $b=t\wedge n$. It follows that $b\text{…
user71346
  • 4,171
3
votes
1 answer

limiting tangent line is parallel to asymptotic line

For a (infinitely, if necessary) differentiable curve $$ A(t) = (x(t), y(t)) $$ which diverges at $t_0 \in [-\infty,\infty] $, that is $$ \lim_{t \to t_0 } | x(t)^2 + y(t)^2 | =\infty $$ if there is a line $l: ax+by+c=0$ such that $$ \lim_{t \to…
Guldam
  • 934
1 2 3
…
99 100