Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

learn more… | top users | synonyms (1)

2
votes
0answers
465 views

Prove that circle is the only curve which spherical indicatrix coincides with it

The task is to prove that a space curve and its spherical indicatrix of tangents coincide if and only if the curve is a circle. Def. As a point moves along a space curve C envision a unit vector t ...
2
votes
0answers
45 views

FLRW metrics (isotropic and homogeneous space)

Consider a spacetime with metric $$ ds^2 = -dt^2 + a^2(t)d\Omega_k^2, \quad k=0,\pm1$$ where $a(t)$ is any regular function and $d\Omega_k^2$ is the 3-dimensional metric of the 3-sphere $S^3$, if $...
2
votes
0answers
64 views

Help with local coordinates computations-differential forms

I am reading a lecture notes on differential forms and tangent bundles on complex manifolds and I got stuck on one line where the author does not explain how he did the computation: Let $\...
2
votes
0answers
95 views

Definition of contact metric structure

I know this is a rather stupid question, but I still need to ask (and I am a physics student, so please excuse me using components): In Blair's book and many other litereatures, the definition of a ...
2
votes
0answers
46 views

On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
2
votes
0answers
184 views

norm of tangent to geodesic is constant

How do you prove that $g(T, T)$ is constant along a geodesic, where $g$ is a metric and $T$ is the tangent vector to the geodesic?
2
votes
0answers
147 views

Definition of a tangent space

Today we defined a tangent space similar to the description here: enter link description here My problem is the following: Why do we need to refer to charts in this case? I mean, would it not be ...
2
votes
0answers
169 views

Relation between exponential map and parallel transport

I'm starting to learn Riemannian geometry and have a question. Let $\mathcal{M}$ be a Riemannian manifold, $p \in \mathcal{M};\ \tau_{p}^{q}$ be a parallel transport from $p$ to $q$ and $\...
2
votes
0answers
84 views

Why are geodesics preserved by the quotient with the isometry group $M/G$?

I'm trying to prove that if $\langle M,g\rangle$ is a riemannian manifold and $G = Isom(M)$ acts properly discontinuous on $M$, then a geodesic $c$ is send to another geodesic by the map $\pi: M \...
2
votes
0answers
72 views

Show that the Mod 2 Linking Number is well defined.

If $X$ and $Y$ are compact, boundaryless smooth manifolds, $f\colon X \to \mathbb R^n$ and $g\colon Y \to \mathbb R^n$ with $f(X) \cap g(Y) = \emptyset$, and with $\dim (X) +\dim (Y) =n-1$. Define ...
2
votes
0answers
47 views

show that $g\circ f :S_1 \to S_3$ be also smooth and $d_p(g\circ f)=(d_{f(p)}g)\circ (d_pf)$

proposition: Let $S_1$, $S_2$, $S_3$ be 3-regular surfaces and $f:S_1 \to S_2$ and $g: S_2 \to S_3$ be smooth maps. Then show that $g\circ f :S_1 \to S_3$ be also smooth and $$d_p(g\circ f)=(d_{f(p)}...
2
votes
0answers
200 views

$k$-jets of sections of a vector bundle..

I need some help for establishing a connection between two definitions of $k$-jets: Algebraic Definition: Let $E\rightarrow M$ be a smooth vector bundle and define the ideal of $\Gamma(E)$: $$I_p(M):...
2
votes
0answers
57 views

Global conformal equivalence of two regions of Minkowski spacetime

I am wondering whether the region $H:=\{(t,x) : x^2-t^2<1\}$ of $(1+1)$-dimensional Minkowski spacetime, equipped with the restriction $g_H$ of the standard Minkowski metric $g=-\mathrm{d}t \otimes ...
2
votes
0answers
44 views

Comparing the norm of a trace of a curvature tensor with the full norm

Let $V$ and $E$ be complex vector spaces of dimensions $n$ and $r$, equipped with hermitian inner products $\omega$ and $h$ respectively. Let $R$ be a curvature-type tensor, that is an element of $\...
2
votes
0answers
144 views

Vector fields on smooth manifolds and Lie algebras

I'm currently studying differential geometry on smooth manifolds using differential forms and I'm trying to apply it to what I have learned earlier about Lie groups, but something doesn't seem to ...
2
votes
0answers
179 views

Local trivializations for orthonormal frame bundle

Let $(E,\pi, M)$ be a real vector bundle of Rank $N$. Then one can define its frame bundle $GL(E)$ as follows: $GL(E)_x:=\{\text{ordered bases of }E_x\}$ (for $x\in M$). $GL(E):=\bigcup_{x\in M} GL(...
2
votes
0answers
64 views

Group generated by several vector fields

I have two (or more) smooth and integrable vector fields $v,w$ on a smooth manifold $M$. Each generates a flow map $\Phi_v$,$\Phi_w$ that forms a single parameter Lie group of diffeomorphisms. Let's ...
2
votes
0answers
156 views

Whitney Embedding theorem for manifold with boundary in Lee's Introduction to smooth manifolds(2nd Edition)

I am reading through the new edition of Lee's book and I am stuck by the proof of Theorem 6.15. When passing to a non-compact manifold, the author begin by defining several sub-levelsets and claim ...
2
votes
0answers
323 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha \...
2
votes
0answers
59 views

Continuity of point wise orientation

Prove that $[(X_1,...,X_n)]$ on a manifold $M$ is continuous if and only if every point $p$ in $M$ has a coordinate neighborhood $(U,\phi) = (U,x^1,...,x^n)$ such that for all $ q \in U$, the ...
2
votes
0answers
601 views

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 ...
2
votes
0answers
59 views

Finding an isometry given the distance between points

If $A,B,C,D \in \mathbb{R}^n$, such that $d(A,B)=d(C,D)$, then exists an isometry $f:\mathbb{R}^n\rightarrow \mathbb{R}^n $, such that $f(A)=C, f(B)=D$. Thanks a lot for the help.
2
votes
0answers
91 views

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 ...
2
votes
0answers
77 views

minimise the total distance to a hyperbolic curve from two fixed points

The point $C$ moves along the hyperbolic curve which is given by $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$. The distances $d_{0}$ and $d_{1}$ in from $A$ to $C$ and $B$ to $C$ respectively. ...
2
votes
0answers
145 views

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 $x$...
2
votes
0answers
144 views

Legendrian Isotopy of Knots can be extended to an ambient Contact Isotopy

I am attempting to understand a proof that an isotopy of two Legendrian knots $L_0$ and $L_1$ in a closed contact manifold (M,$\xi$) can be extended to an contact isotopy $\phi$ of M such that $\phi_0 ...
2
votes
0answers
92 views

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 ...
2
votes
0answers
156 views

Maximal submanifold

I wonder if the notion of "maximal submanifold" exists or is relevant? I'm surprised because I found pretty much nothing about it on the web (after a quick search). The definition, which seems ...
2
votes
0answers
307 views

Construction of line bundles on the flag variety

Edit: The following is phrased in terms of algebraic geometry, but can be thought of analytically as well. Hence I added some tags... I am a bit confused about the subject in the title. For ...
2
votes
0answers
51 views

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(\...
2
votes
0answers
459 views

pullback and pushforward examples

Where can I find some simple examples of pullbacks and pushforwards between manifolds. Specifically examples that show the details of the computations.
2
votes
0answers
44 views

Sufficient statistics and isometries

Let $(M,g)$ be an infinite dimensional statistical manifold with the Fisher information metric $g$. Is it true that any isometry on this manifold must correspond to a sufficient statistic?
2
votes
0answers
116 views

Problem about finding Gaussian curvature

The problem is about Gaussian Curvature When we define $M=r(u,v)$ be the surface parametrized by $(u,v)$ in $\mathbb{R}^3$ and $N$ be a unit normal vector of $M$. In my lecture we denote Gaussian ...
2
votes
0answers
426 views

Horizontal and vertical tangent space of Orthogonal group

We know for the orthogonal group n-by-n orthogonal matrices, the tangents are given by $X^T\Delta + \Delta^TX = 0$ where $\Delta$ is the tangent. Now I was reading about the vertical and horizontal ...
2
votes
0answers
529 views

local parametrization of regular surface

I am doing excercises of Do Carmo's dg of curves and surfaces Chapter 2.2 and need some help with the following excercise: Show that the set $S=\{(x,y,z)\in R^3;z=x^2-y^2\}$ a regular surface and ...
2
votes
0answers
206 views

Is this enough to show that this map has constant rank?

My question is related to this question I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that $f\vert_A=id:A\rightarrow A$. Then $A$ is ...
2
votes
0answers
81 views

Vector field from Lamination.

Let $S$ be a smooth closed (i.e. compact without boundary) surface. A geodesic lamination on $S$ is a nonempty closed subset of $S$ which is a disjoint union of geodesics. Suppose $\alpha$ is a ...
2
votes
0answers
62 views

Problem reduced to analyzing solutions of a family of nonlinear systems of equations

This was posted on mathoverflow about two weeks ago and I got no response so I'm asking here in case anyone has any ideas. Original post is here. I was able to reduce a research problem relating to ...
2
votes
0answers
125 views

$\deg(f)\neq 0\Rightarrow f$ is surjective

How can I prove the following statement? Let $f:M\rightarrow N$ be a smooth map between closed, connected, oriented manifolds of the same dimension. If $\deg(f)\neq 0$, then $f$ is surjective. ...
2
votes
0answers
904 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
2
votes
0answers
132 views

Torsion of fundamental group is abelian

On Riemannian manifold with Ricci curvature bounded below (For example, flat torus. It has ${\bf Z}^2$ as fundamental group, which is nilpotent (=almost abelian). In fact Ricci curvature bouned ...
2
votes
0answers
95 views

Momentum map and equivariance

I am reading an article in which I do not understand some equivariance property about the momentum map. Let $G$ be a Lie group acting on a manifold $Q$. The action is denoted $(g,q) \, \mapsto \, q \...
2
votes
0answers
115 views

Asymptotics of Green's function of laplacian

Let $ \Omega $ be a domain in $ \mathbb{R}^2 $ with a Riemannian metric $ g $, and let $\Delta $ be the laplace-beltrami operator induced by the metric, with dirichlet boundary conditions. For $ x,y \...
2
votes
0answers
118 views

Parallel transport on a submanifold

Let $M$ be a Riemannian manifold and $N$ an embedded submanifold of $M$. Now I have a geodesic $c \colon (-a,a) \to N$ with $a>0$ and $c(0)=p$ in $N$, s.t. $c'(0)$ is orthogonal to $T_pN$. Is the ...
2
votes
0answers
227 views

Exact and Closed forms on Manifolds with Boundary

Let $\bar{M}$ be a manifold with boundary and let $M$ be its interior. Is this statement correct? A smooth k-form $\alpha$ on $\bar{M}$ is closed (exact) if and only if its restriction to $M$, i.e. $\...
2
votes
0answers
32 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y \...
2
votes
0answers
83 views

angle sum for triangle on helicoid

Given the helicoid $$ \boldsymbol{r} = (u\sin v, u\cos v,v)$$ in three-dimensional Euclidean space, consider the triangle $T$ defined by $$ 0 \leq u \leq \sinh v, \qquad 0 \leq v \leq v_0.$$ The ...
2
votes
0answers
97 views

For submersion and submetry,why can we lift a geodesic “horizontally” to a geodesic?

A map $\sigma:X\to Y$ between locally compact complete inner metric spaces is called a submetry if $\sigma(B_r(p))=B_r(\sigma(p))$ for all $r>0$ and $p\in X$. Why is that a geodesic in $Y$ can be ...
2
votes
0answers
158 views

Commutativity of two vector fields. Prooving that $\frac{d}{dt}\mid_{t=0}\alpha (t)=[X,Y]_{p}$

If $X$ and $Y$ are smooth vector fields with flows $\phi^{X}$ and $\phi^{Y}$, then starting at some $p\in M$, if we flow with $X$ for time $\sqrt t$, then flow with $Y$ for time $\sqrt t$ and then ...
2
votes
0answers
184 views
+100

What is the torsion and curvature in modern language?

I think it can not be the torsion and curvature in the connection context, for these two are anti-symmetric in two variables, so that they must vanish on a one-dimensional space (tangent space of a ...