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 ...

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116 views

Is the following method of determining a curve from its normal vector valid?

One of my homework problems asks us to show that the curvature and the torsion of a regular parameterised curve with non-zero torsion everywhere are uniquely determined, when we already have the ...
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217 views

Plane curve: orhogonal projection and closest points

I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28: "Let two curves $C_1$ and $C_2$ have a regular point $P$ ...
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66 views

show that subset of complex projective space is a submanifold

Let n, m $\in \mathbb{N}$. I'm trying to show that $M(n,m) = \{[z_0 : z_1 : … : z_n] \in \mathbb{C}P^n | \sum^{n}_{i=0} z^m_i = 0\}$ is a submanifold and codim(M(n,m))=2. My idea was to use ...
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52 views

Prove that $X$ parametrizes a regular surface $M$ in $\mathbb{R}^3$ and determine for which values $p$ the curve $y$ is geodesic on $M$.

Real valued functions $f,g:\mathbb{R}_+ \rightarrow \mathbb{R}$ are $f(u) = e^{-u}$ and $g(u) = \int \sqrt{(1-e^{-2t}} dt$. Given $X:\mathbb{R}_+ \rightarrow \mathbb{R}^2$ with $X(u,v) = [f(u) \cos ...
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67 views

Isomorphism between tensor product and set of all tensors

I'm new to tensor and quite confused. First of all, could anyone provide any friendly reference about tensors which explains the idea behind those boring-looking definition? Then is my problem. Let ...
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80 views

Is there an action functional whose critical points are the geodesics of a arbitrary connection on TM?

Geodesics of the Levi-Civita connection may be defined as the critical points of the action functional $S[\gamma]=\int \lvert\dot{\gamma}\rvert\,dt$ (or square it, if you like). The Euler-Lagrange ...
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72 views

Problems about dual map, cotangent bundle.

I have no idea what dual space and dual map are, so have much trouble understanding cotangent bundle when reading Lee's book. First of all, can anyone give me a introduction what the dual map and ...
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270 views

Extension of a smooth function on a set of manifold

I encountered the following proposition: If a function is smooth on an arbitrary set $S\in M$, where $M$ is a smooth manifold, then it has a smooth extension to an open set containing $S$. It seems ...
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236 views

What is a direct proof of isoperimetric inequality?

What is a direct proof of isoperimetric inequality? In another word, i am asking for a proof that the circle has the maximum area compare to other geometric shape with the same circumstance but ...
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97 views

Lie group GL(n,R) and the determinant map

Let $d : GL(n,R) \rightarrow R$ be the determinant map. I don't know how to prove that if the map $d* : T_{I_n}GL(n,R) \rightarrow T_1R$ is surjective, then the map $d* : T_AGL(n,R) \rightarrow T_1R$ ...
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97 views

Submanifold of $\mathbb{R}^4$

In the space of $2\times 2$ matrices, find explicitly the sets of matrices with 1)a single zero eigenvalue, 2) a pair of pure imaginary eigenvalues. Show that each set is a submanifold of ...
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31 views

Matrices of forms seen as sections of a vector bundle

Given a vector bundle $\pi: E \rightarrow M$ , what are the sections of $\Omega^p(\operatorname{End} E)$? are they just matrices whose entries are $p$-forms?
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100 views

What is the geodesic equation on $\mathbb{S}^{n}$?

Suppose $\gamma: \mathbb{R}\rightarrow \mathbb{S}^{n}$ is a smooth curve. Let $\gamma(t)=(x^{1}(t)...x^{n+1}(t))$. Let $\mathbb{D}^{n}$ be embedded into $\mathbb{R}^{n+1}$ by viewing ...
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127 views

Homogeneity lemma

I am studying Homegeneity lemma. I am not understanding the following paragraph: Given any fixed unit vector $c \in S^n$, consider the differential equations $\frac{dx_i}{dt} = c ...
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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) ...
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94 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 ...
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53 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) := ...
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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?
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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 ...
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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: ...
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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 ...
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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 ...
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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 ...
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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), ...
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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 ...
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217 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 ...
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451 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 ...
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185 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 ...
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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 ...
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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 ...
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166 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 ...
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142 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 ...
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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 ...
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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 $r$) ...
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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 ...
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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 ...
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93 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 ...
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162 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 ...
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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 ...
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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 ...
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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} : ...
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145 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?
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226 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 ...
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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,& ...
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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, ...
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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 .
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240 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 ...
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45 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?
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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 $f(x) \ne f(y)$?
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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, ...