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)

1
vote
0answers
160 views

Closest point projection of manifolds in Banach spaces

Suppose that $Y$ is an infinite dimensional Banach space, with an embedded finite dimensional compact submanifold $X$. It is well known (cfr Lang's Differential and Riemannian Manifolds, Thm. 5.1) ...
1
vote
0answers
100 views

Neighborhood Retraction of Boundary

Here is the problem: If $M$ is a manifold with boundary, then find a retraction $r:U \rightarrow \partial M$ where $U$ is a neighborhood of $\partial M$. I realize that the collar neighborhood ...
1
vote
0answers
55 views

A K-invariant submanifold of G-manifold and fundamental vector fields

Let a (connected) Lie group $G$ act on $M$. Assume that the action is locally free. (In other words, if the fundamental vector field of $X \in \mathrm{Lie(G)}$ $$ \underline{X}(p) := ...
1
vote
0answers
92 views

extension of the exterior derivative

Is it true that since the smooth forms are dense in the $L^2$ and $H^{1,2}$ sections of forms, we can extend the exterior derivative $d$ and its adjoint $d^*$ to this spaces?
1
vote
0answers
66 views

Pullback of conformal killing field via conformal map

This is my first time posting on this forum, so to start with, it's good to meet you all and thanks in advance for the help! My question is as follows. Suppose I have two semi-riemannian manifolds of ...
1
vote
0answers
103 views

Given a vector field all of whose integral curves are closed, is the period a smooth function?

I am reading about the energy-period relation for Hamiltonian Systems. In Weinstein's formulation (cf. Abraham, Marsden, Foundations of Mechanics 2nd Ed, page 198) this relation amounts to: ...
1
vote
0answers
89 views

Analog of a tubular neighborhood for an embedded wedge sum

If you have some embedding of a path connected topological space wedge of spheres $N$ into a compact simply connected smooth $n$ manifold $M$ (like a sphere for example), then is there some kind of ...
1
vote
0answers
199 views

Closed Geodesics as minimisers of action functional

Suppose I have a Riemannian surface $(M,g)$. It's clear that closed geodesics are critical points of the length functional $l(\gamma)=\int\left|\gamma(t)^{\prime}\right|_{g(\gamma(t))}dt$ over the ...
1
vote
0answers
46 views

why do i need convexity of the set of positive definite hermitian matrices for hermitian structure on vector bundle?

Consider a Riemann surface $\Sigma$ and a holomorphic vector bundle $\mathcal{E} \to \Sigma$ of rank $N$. I just came across the remark that in order to endow $\mathcal{E}$ with a Hermitian metric it ...
1
vote
0answers
87 views

Manifold contains a totally geodesic closed hypersurface

Let $(M^n,g)$ be a closed simply-connected positively curved manifold. Show that if $M$ contains a totally geodesic closed hypersurface (i.e., the second fndamental form or shape operator is zero), ...
1
vote
0answers
93 views

The Implicit Function Theorem and open sets with regular boundary

Let $\rho :\mathbb{R}^N\longrightarrow\mathbb{R}$ be a continuously differentiable function such that $\rho (x) = 0 \Rightarrow d\rho (x) \not = 0$ for all $x\in\mathbb{R}^N$. Suppose $\Omega ...
1
vote
0answers
223 views

Difference between Geodesic and principal lines of curvature

As much as i understand. Geodesic line of curvature is a line on the surface such that the projection on the tangent plane of it's curvature vector is 0 at every point. The lines of curavture are ...
1
vote
0answers
456 views

Question about Bertrand Curve

A regular curve $C$ in $\mathbb R^3$ is called a Bertrand Curve, if there exists a diffeomorphism $f:C \to D$ from $C$ onto a different regular curve $D$ in $\mathbb R^3$ such that ...
1
vote
0answers
186 views

differentiating under the integral sign

Suppose I have a moving curve $\alpha:I\times[0,T] \to \mathbb{R}^n.$ Its length is $$\int_\alpha ds = \int_I |\alpha_x|dx.$$ If I want to find the time derivative of this, I guess I differentiate ...
1
vote
0answers
87 views

$C_0$ Convergence of Metrics under Ricci Flow when $\chi(M) = 0$.

I am a beginning graduate student, and I am trying to learn basic aspects of Ricci Flow via the uniformization theorem for compact surfaces. I am reading Chow and Knopf's introductory book as well as ...
1
vote
0answers
63 views

Set of points where an application has rank $m$ is a smooth manifold.

Can someone help me with this problem? I have a $C^1$ function $G\colon\mathbb{R}^n\rightarrow \mathbb{R}^m$, where $k=n-m> 0$. If $M$ is the set of points $x\in G^{-1}(0)$ such that $(DG)_x$ has ...
1
vote
0answers
170 views

The Closed disc $D$ is a manifold with boundary

It is obvious geometrically, but how one proves with a few words, analytically, the statement above?Additionally, if one has a smaller open disc $D_\epsilon$ of radius $\epsilon$ centered at a point ...
1
vote
0answers
144 views

Why topological stratification is useful?

My main focus is on the applications of stratification in complex/abstract Algebraic geometry especially, from the scheme-theoretic viewpoint and (Added) Moduli spaces. I have a vague feeling that ...
1
vote
0answers
60 views

Relations between metric on H and the disc model

In the book "Modular Forms" by Miyake one finds the definition of some obscure 'thing'. He calls it a metric on $\mathbb{H} = \{z \in \mathbb{C} : \operatorname{Im}(z) > 0 \}$. The following is ...
1
vote
0answers
131 views

Energy displacement of a cylinder is at most $\pi r^2$.

I want to show that the energy displacement of $Z^{2n}(r)$ is at most $\pi r^2$. In the textbook of Mcduff and Salamon they write that I should identify the two dimensional ball (with radius $r$) ...
1
vote
0answers
162 views

Isometry from a surface to itself.

We are talking about surfaces in $\mathbb{R}^3$ here. I know that not every isometry from a surface to another is a congruence. But what about isometries from a surface to itself? Can someone give ...
1
vote
0answers
61 views

How to find the boundary of a $\mathcal{C}^1$ manifold?

Let $U$ be a bounded open convex set of $\mathbb{R}^d$, and $\Phi$ a differentiable map from $U\times \mathbb{R}$ to $\mathbb{R}^d$. The object of interest for me is $R=\Phi(U\times \mathbb{R})$. The ...
1
vote
0answers
94 views

Different definitions of the complex Grassmannian

I have come across two different definitions of what I suspect is the same object. Both are called the complex Grassmannian: 1: $U(n)/U(k)\times U(n-k)$ 2: $SU(n)/(S(U(k)\times U(n-k))$ What is the ...
1
vote
0answers
165 views

The connected Lie subgroups of a n-dimensional torus.

I am stuck on the following question Let $T^n$ be a $n$-dimensional torus. What are the connected Lie subgroups of $T^n$? Which of these are closed subgroups? Does anyone have any ideas on how to ...
1
vote
0answers
182 views

Hopf-Rinow theorem

If $(M,g)$ is a riemannian manifold. $M$ is complete(geodesically) then any two point can be joined by a geodesic. Geodesic($\gamma(t))$ is smooth curve such that ...
1
vote
0answers
119 views

$S^1 \times S^1 \ldots \times S^1 $ is diffeomorphic to $\mathbb{T}^n$

how do I show that $S^1 \times S^1 \ldots \times S^1 $ is diffeomorphic to $\mathbb{T}^n$ as manifolds? Where $S^1 \times S^1 \ldots \times S^1 $ has the natural differential structure of a product ...
1
vote
0answers
167 views

Riemannian connections: how to understand $\overline{\triangledown}_X \langle Y,Z \rangle$

I have a question regarding a comment in Lee's book "Riemannian Geometry - an Introduction to Curvature". On page 52, Lee introduces the Euclidean Connection as the map $\overline{\triangledown} : ...
1
vote
0answers
146 views

Differential map between smooth manifolds is smooth

Given a smooth map $f:M\to N$ between smooth manifolds how do you show that the differential map $df:TM\to TN$ is smooth?
1
vote
0answers
227 views

Proof of Gauss's lemma in Riemannian geometry

In the proof of Gauss's lemma here, there is a step $\displaystyle\lim_{t\to0}\frac{\partial f}{\partial s}(0,t)=\lim_{t\to0}T_{tv}\ \exp_p(tw_N)=0$ However, the limit seems meaningless (unless ...
1
vote
0answers
80 views

Is this function on the surface smooth?

Consider the following functions $F_{ij}:S\subset{\mathbb R}^3\to{\mathbb R}$, $$ F_{ij}(y) = \begin{cases} \frac{(y_i-x_i)(y_j-x_j)(y-x)\cdot n(y)}{|y-x|^3},&y\neq x; \\ 0,& ...
1
vote
0answers
86 views

Isoperimetric inequalities with relative perimeter

It is a well known result that if $\Omega\subset \Bbb{R}^N$ is an open set, with regular boundary (smooth, or Lipschitz) then the problem $$ \min_{E \subset \Omega |E|=c} Per(E)$$ has a solution, ...
1
vote
0answers
187 views

$SU(2)$ is a covering space of $SO(3)$.

The method of topology is very clear.Then there's a question asking to use adjoint representation of lie group $SU(2)$ $(\operatorname{adj}:SU(2)\to GL(su(2)))$to prove this. I can't solve this .
1
vote
0answers
245 views

Gaussian Curvature

Can anyone see the connection between the Gaussian curvature of the ellipsoid $x^2+y^2+az^2-1=0$ where $a>0$ and the integral $\int_0^1  {1\over (1+(a-1)w^2)^{3\over 2}}dw$? I am guessing ...
1
vote
0answers
47 views

Characterizing surfaces in $R^3$ in which every point is an umbilic point

How can it be shown that the only such surfaces are spheres or planes?
1
vote
0answers
51 views

Function seperating points

If $M$ is a Hausdorff $n$-manifold (without further assumption like paracompactness), given $x,y$ in $M$ is there a smooth function $f$ such that $f(x) \ne f(y)$?
1
vote
0answers
298 views

Deriving Euler-Lagrange equation and natural boundary conditions for two-phase piecewise $H^1$ Mumford-Shah model

Let $\Omega \subset \mathbb{R}^2$ be a bounded open domain. Derive the Euler-Lagrange equations and the natural boundary conditions for the two-phase piecewise $H^1$ Mumford- Shah model: $$J(\varphi, ...
1
vote
0answers
123 views

First-order derivatives in differential forms calculus

Let $d$ denote the Cartan differential, and let $\delta$ denote the codifferential. The underlying domain is not important for what follows. The canonical generalization of the Laplace-operator ...
1
vote
0answers
58 views

Cartan immersion of symmetric spaces.

I got stuck on some fact concerning the Cartan immersion of symmetric spaces: Let $M$ be a Riemann symmetric space associated to the Lie group $G$ with involutive automorphism $\sigma$. Let $G$ act ...
1
vote
0answers
118 views

Help on Einstein Summation

I am not sure how to interpret the following expression with regard to the Einstein summation convention \begin{equation} g^{ab}(\partial_c \Gamma^c_{ab} - \partial_b \Gamma^c_{ac}) \end{equation} ...
1
vote
0answers
96 views

Distinct metrics on a manifold

I'm trying to understand basic differential geometry (my background is in mathematical logic), and I'm having a bit of difficulty with a basic point. Frequently we want to consider the set of metrics ...
1
vote
0answers
662 views

Relationships between curvature, torsion, unit tangent vector, and binormal vector of a curve

Homework has already been collected and graded (but no explanation given) for these problems. I'm curious how to approach the problem. Assume that the vector space we're in is $\Re^{3}$. Prove that ...
1
vote
0answers
97 views

Another time on jets and composition

Suppose we have four smooth maps between smoot manifolds: $$f: M \rightarrow X$$ $$g: X \rightarrow N$$ $$h: M \rightarrow Y$$ $$i: Y \rightarrow N$$ an the equation on compositions of jets $$j_m(g ...
1
vote
0answers
121 views

Stereographic projections and cross-ratios

Would anybody shed some light on question 2.11 in Wilson's Curved Spaces? The numbers $p,q\in \hat{\mathbb{C}}$ are stereographic projections of points $P,Q$ on the unit sphere. The spherical ...
1
vote
0answers
61 views

Complex cones and a foliation of $\mathbb{P}^3$

1 Take the 3-dimensional complex projective space $\mathbb{P}^3$. Consider the action of the group $SU(2)\times SU(2)$. I have read in physics related articles that these group gives a singular ...
1
vote
0answers
163 views

hermitian distance functions and geometry in complex space

If we have fixed a hermitian positively definite form $h(.,.)$ in complex space $C^n$ and an analytic submanifold $M$ in $C^n$, then we may fix a point outside of $M$, say $P$, and consider distance ...
1
vote
0answers
241 views

Deriving an expression for minimum arc length along a 3D surface between any two points.

Consider a 3D surface, defined by the function $z = f(x, y)$. Assuming the surface is differentiable (no kinks), is there a function that expresses the minimum arc length traced along the surface ...
1
vote
0answers
53 views

Smooth mapping $v \colon [0,1] \to S^{n-1}$

I have a smooth mapping $v \colon [0,1] \to S^{n-1}$ such that for any $u \in S^{n-1}$ exists $t \in [0,1]: v(t)\cdot u = 0$ and $n \geq 3$. So a have an assumption that such a mapping $v(\cdot)$ ...
1
vote
0answers
182 views

Local theory of curves using Taylor expansion

Consider a $C^m$ curve $\gamma$ in $\mathbb{R}^3$ (with $m\geq 3$). Locally, at $0$ for convenience, one can express the curve as $$ \gamma(s) = \gamma(0)+sT+(s^2/2!)kN+(s^3/3!)(\dot{k}N-k^2T+k\tau ...
1
vote
0answers
74 views

Approximation in the Plane of Constant Curvature

In Euclidean space, There is a classic Theorem claims that: The length of every rectifiable curve can be approximated by sufficiently small straight line segment with ends on the curve. Now, The ...
1
vote
0answers
121 views

Vector equation and curvature

Vector equation $r(t)=2\cos(t)\mathbf i + 3\sin(t)\mathbf j\ \ (0 \le t \le 2\pi)$ represents ellipse. I need to find curvature of this ellipse on endpoints of x and y axis that are given with ...