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

If a closed, smooth $m-1$ form, $\omega$ is nonzero at a point, there are local coordinates $x^i$ with $\omega = dx^2 \wedge\cdots \wedge dx^m.$

This is a problem on an old qualifying exam. Let $\omega$ be a smooth, closed $m-1$ form on a smooth $m$-dimensional manifold $M$. If $\omega \neq 0$ at a point $p\in M$ then there is a coordinate system $(x^1,...,x^m)$ on a neighborhood of $p$ in…
Bohring
  • 549
2
votes
0 answers

plane curves and osculating plane

Let $\alpha$ be a curve such that $|\alpha'(s)|=1$ for all $t$ and $k\neq 0$. The tangent vector $\vec T(s)$ and the normal vector $\vec N(s)$ through $\alpha(s)$ span a plane called the osculating plane. Show that $\alpha$ is a plane curve if and…
dinosaur
  • 2,252
2
votes
1 answer

Find isometry group of the Poincaré disc

I cannot show that any isometry of the Poincaré disc $\mathbb{H}^2$ is given by $e^{i\theta} \frac{z + c}{cz + 1}$ where $\theta$ and $c$ are constants. I know that they are indeed isometries, however I don't know how to prove that any isometry is…
user40276
  • 5,283
2
votes
0 answers

Volume forms under diffeomorphisms

I have an exercise and I have no idea how to do it: Let be $U$ and $V$ open sets of $\mathbb{R}^{n}$ and $f:U\rightarrow V$ an orientation-preserving diffeomorphism, then $$f^{*}(\operatorname{vol}_{V}) = \sqrt{\det(g_{ij}(x))}…
CerealKiller
  • 51
2
votes
0 answers

Transforming the measure in $CP^1$ mapping from Riemann sphere to $\mathbb{C}^2$-plane

I would like to know how the measure changes in $CP^1$ mapping from Riemann sphere (2-sphere) to $\mathbb{C}^2$-plane. Let a point on the 2-sphere is given by the vector $\hat{n}(\mathbf{x})=(n^x(\mathbf{x}),n^y(\mathbf{x}),n^z(\mathbf{x}))$ with…
maxr
  • 131
2
votes
0 answers

Metric on the sphere involving tensor product

The metric on the sphere in the $(\theta,\phi)$ is of the form: $$ g_{\theta\theta}=r^2,g_{\theta\phi}=g_{\phi\theta}=0,g_{\phi\phi}=r^2\sin^2\theta $$ When transforming it to the $(x,y)$ coordinate on the plane using stereographic projection, I got…
hxhxhx88
  • 5,257
2
votes
1 answer

nonorientability of the projective plane

Let P be the real projective plane obtained by identifying antipodal points on the unit sphere of $R^3$. How to prove that P is nonorientable in a rigorous and elementary way? I do not want mere intuition. My idea is to consider the closed curve …
noot
  • 71
2
votes
3 answers

Normal and principal curvature

Suppose that $v$ and $w$ are orthogonal unit vectors in $T_p\Sigma.$ Show that $\kappa_p(v)+\kappa_p(w)$ is independent of the specific choice of $v$ and $w$ as long as they are orthogonal. $T_p\Sigma$ is the tangent plane of a surface…
Lays
  • 2,023
2
votes
1 answer

velocity vector on the projective plane

Let P be the projective plane obtained by identifying antipodal points on the unit sphere. Let $\alpha$ be a curve in P. The book I am reading says that $\alpha'(t)$ is the function such that $\alpha'(t)[f]=\frac{d}{dt}f(\alpha(t))$ for every…
noot
  • 757
2
votes
0 answers

Smooth boundary is a smooth manifold

Let $M\subset \mathbb{R}^n$ be a $k$-dimensional manifold and $X\subset M$ a subset. The boundary of $X$ in $M$, denoted by $\partial_M X$, is the set of all $x\in X$ such that each neighborhoud of $x$ contains points in $X$ and $M\setminus X$. I…
student
  • 21
2
votes
1 answer

Principal and normal curvature

I am having trouble understanding the principal curvature and normal curvature. What are the basic intuition or ideas between the two. How are they related? And if given a problem, how can you compute the normal and principal curvature? For…
Lays
  • 2,023
2
votes
1 answer

A better proof for $\mathbf{III}-2H\mathbf{II}+K\mathbf{I}=0$

From here , the equation 7 gives $$\mathbf{III}-2H\mathbf{II}+K\mathbf{I}=0$$ where $\mathbf{III},\mathbf{II},\mathbf{I}$ are third, second and first fundamental forms respectively. $H$ is the mean curvature and $K$ is the Gauss curvature. I can…
Golbez
  • 4,276
2
votes
3 answers

Integral of the gaussian curvature on surface.

Let $S \subset \mathbb{R}^3$ be the surface given by $x^4+y^4+z^4=1$, and let $K$ be its gaussian curvature. Then what is $\int_S K$? First of all, I think finding patch seems hard. Should I consider 8 patches given by $(x,y,z)=(\pm \sqrt{\sin u…
Gobi
  • 7,458
2
votes
0 answers

Why a regular surface could not have boundaries?

I'm reading the differential geometry written by DoCarmo and having trouble when understanding the definition of regular surface. What troubles me is that I could not see why the definition would rule out those case when the surface has boundaries.…
lucille
  • 21
2
votes
1 answer

Why is $f:\mathbb{R}\to S^1$ $f(t)=(\cos(t),\sin(t))$ a local diffeomorphism?

An example in my book says that $f:\mathbb{R}\to S^1$ defined by $f(t)=(\cos(t),\sin(t))$ is a local but not global diffeomorphism. By the inverse function theorem, $f$ is a local diffeomorphism if the determinant of $df_x$ is nonzero. I must be…
PSK
  • 29
1 2 3
…
99 100