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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613 views

Hodge Star Operator

I'm trying to understand the Hodge star operation, but have come across an impasse almost immediately. I have the definition $$(\star \omega)_{a_1\dots a_{n-p}}=\frac{1}{p!}\epsilon_{a_1\dots ...
2
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0answers
67 views

Difeomorphisms and boundary conditions

So I asked on physics.stackexchange, but got no answer, so I'll try here: I am trying to find out how did the authors in this paper (arXiv:0809.4266) found out the general form of the diffeomorphism ...
1
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1answer
56 views

Minimising Length and Energy for Finsler Manifold

Is it true that a minimiser of Finsler energy is automatically parameterised by arc length? As in the Riemannian case. Is there a reference for this fact?
8
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1answer
128 views

Hyperbolic diameter of Amsler's surface

I've recently learned about Amsler's surface, a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of ...
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0answers
74 views

Ricci tensor in complex space forms

Let \begin{align} \notag f:M^{2n}\to CQ_{c}^{N} \end{align} be an isometric immersion of a Kaehler manifold into a complex space form. We consider an orthonormal basis $Y=X_{1},..,X_{2n}$ and then ...
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138 views

Integration on manifold, pullback

Define the maps $F_\pm:B^2\rightarrow S^2$, $(x,y)\rightarrow(x,y,\pm\sqrt{1-x^2-y^2})$. Prove that for any $\omega\in\Omega^2(S^2)$: $$\int_{S^2}\omega=\int_{B^2}F_+^*\omega-\int_{B_2}F_-^*\omega$$ ...
6
votes
1answer
237 views

Gauss-Bonnet theorem for spheres that almost look like a torus

[Corrected due to Jason's answer.] Imagine a torus and a flat disk fitting in the middle of its "hole" (a doughnut with a membrane in the middle). Cut the torus at its inner equator, duplicate the ...
5
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1answer
119 views

For a differentiable map $\Phi$ between manifolds $M$ and $W$, what is $d\Phi?$ (Aubin)

I can't understand a passage from A Course in Differential Geometry by T. Aubin. First, there is Definition 2.6., which I posted in this question. And now there's this: $(\Phi_*)_P$ is nothing ...
4
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3answers
287 views

Does the gradient always point outward of a level surface?

Let $f:\mathbb{R}^n\to \mathbb{R}$ be a differentiable function, with $a\in \mathbb{R}$ a regular value of $f$. Let $M=f^{-1}((-\infty,a])$. Then $M$ is an $n$-manifold with boundary, whose boundary ...
6
votes
2answers
216 views

Embedded surface in $\mathbb{R}^3$

Let $U \subseteq \mathbb{R}^2$ be an open set and let $\sigma : U \rightarrow \mathbb{R}^3$ be a parametrization of an oriented surface $S$ embedded in $\mathbb{R}^3$ whose unit normal in $\sigma ...
2
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0answers
53 views

Curvature estimates in Hodge theory [closed]

I was wondering what is the importance of estimating the curvature of the Hodge Bundles in Hodge theory. I mean is there any real advantage of doing that? And what about curvature of periods domain? ...
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3answers
207 views

What is a tangent bundle? (Aubin)

Here's what I read in A Course in Differential Geometry by Thierry Aubin. 2.5. Definition. The tangent bundle $T(M)$ is $\bigcup_{P\in M} T_P(M).$ And then 2.6. Definition. Let $\Phi$ be a ...
2
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0answers
76 views

curvature of a general curve

Let $\gamma : I \to \mathbb{R}^2$ be a smooth immersed curve, i.e. $\mathcal{C}^\infty$ and $\frac{d\gamma}{dt}\neq 0$ on $I$. I have found the following formula for the curvature $\kappa$ of ...
2
votes
1answer
93 views

tangential and normal projection of a vector in the ambient vector field of a sphere

I'm having unexpected trouble to perform this computation: Let $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=3\}$ and $v_p = (1,0,0)_{(1,1,1)}$ be a vector from the ambient vector field on $M$. How do I ...
4
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0answers
55 views

non-constant curve c with $\frac{\mathrm{d}c}{\mathrm{d}t}= \lambda\frac{\mathrm{d}^2c}{\mathrm{d}t^2}$

I don't know how to solve the following problem and would appreciate some help. Let $M$ be a submanifold of euclidean space and $c:[a,b] \to M$ a non-constant curve, such that the velocity field of ...
5
votes
1answer
202 views

showing zero curvature implies a line

How can I show that a given (not necessarily unit-speed) parametrization $\gamma(t)$ of a curve in $\mathbb{R}^3$ which exhibits zero curvature is a line ? What I know is that zero curvature means ...
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1answer
129 views

curvature of space curve

I am slightly confused by the following curve $\gamma(t) = (e^t,0,0)$ in $\mathbb{R}^3$. Its curvature, defined as $$ \kappa(t) = \frac{\|\dot \gamma(t) \times \ddot \gamma(t)\|}{\|\dot \gamma(t)\|^3} ...
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2answers
261 views

Why is the tangent space to a point $p$ of $D\subset \mathbb{R}^n$ isomorphic to $\mathbb{R}^n$

I'm having problems with understanding why is it the case. Suppose $D\subset\mathbb{R}^n$ is an open, connected subset and for $p\in D$ define the tangent space $T_pD$ to be the set of the velocities ...
2
votes
1answer
94 views

Dimension of graphs (Differential Geometry)

I have a rather basic question about the dimension of a graph mapped by the exponential map. The problem is I can't really visualize it. It starts like that: Let $M$ be a smooth manifold of ...
3
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0answers
165 views

Schoen estimates for stable minimal surfaces

I'm refering to article of R.Schoen 'Estimates for stable minimal surfaces in three dimensional manifolds' (1983). In the first paragraph of the proof of theorem 2 the author wants to apply the ...
2
votes
1answer
137 views

Differential of the exponential map on the sphere

I have a problem understanding how to compute the differential of the exponential map. Concretely I'm struggling with the following concrete case: Let $M$ be the unit sphere and $p=(0,0,1)$ the north ...
4
votes
3answers
740 views

Showing a vector field is tangent to the 2-sphere

How would I go about showing that the vector field $X = x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x}$ is tangent to the unit sphere in $\mathbb{R}^3$? I can see it pictorially but I'm ...
2
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1answer
168 views

Implicit function theorem-show that in a neighbourhood of the point -can be described by a pair of functions

Let $g_1(x,y_1,y_2)$= $x^2(y_1^2+y_2^2)$-5 and $g_2(x,y_1,y_2)$=$(x-y_2)^2$+$y_1^2$-2. Use implicit function theorem to show that in a neighbourhood of the point x=1, $y_1$=-1, $y_2$=2, the curve of ...
3
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2answers
475 views

is this set a regular surface?

I'm reading "Differential Geometry of Curves and Surfaces of Manfredo Docarmo" I'm doing the exercises of the chapter 2. Here is the definition of regular surface that we are following: I have ...
2
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0answers
111 views

Is multivariable calculus synonymous with differential geometry?

Or are they two distinct topics? For instance, Spivak's calculus on manifolds book considered a treatise on multivariable calculus, but concludes with a differential geometry theorem - Stokes' ...
8
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1answer
462 views

Closed not exact form on $\mathbb{R}^n\setminus\{0\}$

I'd like to construct a closed but not exact $n-1$-form $\omega$ on $\mathbb{R}^n\setminus\{0\}$ in analogy to the winding form: $$\frac{x~dy-y~dx}{x^2+y^2}$$ I think something like ...
2
votes
1answer
106 views

Independence of coordinate charts in the definition of the order of a pole of a meromorphic 1-form on a Riemann surface

I am currently reading Forster. Let $X$ be a Riemann suface, and $Y$ an open subset of $X$. Assume we have a meromorphic 1-form $\omega \in \Omega(Y \setminus \{a\})$ with a pole at $a \in Y$. One ...
1
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1answer
54 views

Show that $(f\circ c_{1})'(0)=(f\circ c_{2})'(0)$

I am trying to solve a problem in Jeffrey Lee's book "Manifolds and Differential Geometry" Ex 2.5: Let $c_{1}$ and $c_{2}$ be smooth curves mapping into a smooth manifold $M$, each with open interval ...
0
votes
1answer
85 views

$Sp(V)$ acts transitively on $V^*-\{0\}$ where $\Omega$ here is symplectic 2 form

Let $\dim(V)=6$. Show that $Sp(V,\Omega)$ acts transitively on $V^*-\{0\}$, where $\Omega$ here is a symplectic 2 form on $V$. ($V^*$ here is algebraic dual of $V$)
2
votes
1answer
209 views

Tangent cone to a subset of $\mathbb{R}^3$

Well, I have the set $X=\{(x,y,z) \in \mathbb{R}^3 | 3x^2+2x^3+y^2+z^2=1\}$ How can I calculate the tangent cone at the point $(-1,0,0)$ ? What are the standard ways to calculate the tangent cone to ...
5
votes
1answer
231 views

a doubt on manifold with boundary, critical point, space of jets etc

could any one explain me the following paragraph by a simple example? "a manifold with boundary is understood to be a smooth (real or complex) manifold with a fixed smooth hypersurface. Two functions ...
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0answers
96 views

Existence of Spin Group

"In mathematics the spin group Spin(n) 1[2] is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie groups As a Lie group Spin(n) therefore ...
2
votes
1answer
258 views

Why is this not a proof of Invariance of Domain?

We know that if $f:K \to X$ is continuous and injective, $K$ is compact, and $X$ is Hausdorff, then $f$ is a homeomorphism $K \cong f(K)$. So suppose $f:U \to \mathbb{R}^n$ is continuous and ...
43
votes
1answer
2k views

What is the solution to Nash's problem presented in “A Beautiful Mind”?

I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve ...
4
votes
1answer
57 views

Describing the algebra of functions on $S^2$

Chapter 2 of the book "Elements of Noncommutative Geometry" claims that the $C^*$-algebra of functions on $S^2$ can be described as an algebra with 3 generators a,b,c all with norm 1, where $a,b$ are ...
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votes
1answer
70 views

Is the set of points in $\mathbb R^n$ with $\sum_{j=1}^n x_j^k = 0$ a submanifold?

Consider the set $$A:= \{x\in \mathbb R^n :\sum_{j=1}^n x_j^k = 0\}$$ for $k$ an odd integer. Is this a submanifold of $\mathbb R^n$ for every $n$? For $n=1$, it is just 0; for $n = 2$, it is the ...
3
votes
3answers
73 views

Definition of a tensor field

Could anybody explain to me the following: If $$T_{ij}=\nabla_i V_j-\nabla_j V_i$$ where $V_i$ is a covector field and $\nabla_i$ is the covariant derivative, then $T_{ij}$ is a tensor field. ...
4
votes
1answer
418 views

good problem book in differential geometry

What are the books in Differential Geometry with good collection of problems. At present I am having John M.Lee's Riemannian Manifolds,Kobayashi Nomizu's Foundations of Differential Geometry. I ...
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0answers
403 views

Curvature of Hyperbolic Space

I'm trying to prove that hyperbolic space has constant sectional curvature $-1$, but keep running into difficulties. Could someone show me a way out? I've been given the metric ...
1
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1answer
97 views

Complete non compact riemannian manifolds and cylinders

Let M be a complete oriented non compact Riemannian manifold with universal covering space (endowed with covering metric) conformally equivalent to the complex plane (with euclidean metric). Then M is ...
2
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1answer
60 views

Angle sum of rays

The sum of angles of a triangle depends on the curvature of the surface and can deviate from $\pi$. What about the sum of angles between successive lines emanating from a given point P? Can it deviate ...
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2answers
130 views

If $\operatorname{dim} M > \operatorname{dim} N$, is there an injective smooth map $M\to N$?

Let $m>n$ and suppose $M$ is a smooth $m$-manifold, $N$ is a smooth $n$-manifold. Can there be an injective smooth map $f:M\to N$?
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1answer
206 views

a neighbourhood of identity U generates G where g is a connected lie group

Let G be a connected Lie group and U any neighbourhood of the identity element. How to prove that U generates G.
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1answer
75 views

Are concepts and properties studied in a category all preserved by morphisms?

When study a category, are we only interested in those concepts and properties preserved by the morphisms, not those which cannot be preserved? For example, in Terry Tao's blog We say that one ...
3
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1answer
76 views

Differentials and second order derivative

I have a question from my tutorials and I don't know how to start... Let $U$ be open in $\mathbb{R}^{n}$ and let $f:U\rightarrow \mathbb{R}$ a $C^{2}$ function. Let $p$ be a point in $U$ where ...
6
votes
2answers
225 views

Detail about the definition of orientability

I am a bit struggling with one of the many definitions of orientability. In what follows $M$ will always denote a smooth, connected manifold, $T_{m}M$ will be the tangent space at $m$, ...
6
votes
3answers
344 views

Exterior algebra of a vector bundle

Associated to any vector space $V$ is its exterior algebra $\Lambda(V)$ which has the direct sum decomposition $\Lambda(V) = \bigoplus_{i=0}^n\Lambda^i(V)$ where $n = \dim V$. My first interaction ...
3
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1answer
203 views

How to compute the index of a given vector field on a triangular mesh

Suppose that I have a triangular mesh (discrete surface composed of triangles). Now, I have been given a vector field (one vector with each triangle, tangential, unit length, so can be represented by ...
2
votes
1answer
141 views

Tangent space as the dual of an ideal quotient

I'm just trying to understand better this way of seeing the tangent space. Given a manifold $M$, it's possible to define the tangent space as $(\mathfrak{I}/\mathfrak{I^2})^*$ , being $\mathfrak{I} = ...
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2answers
1k views

How to make a formula that will interpolate a curved line graph?

In this curved line graph, I need to be able to make a formula that can tell me the interpolated value at any point on the curved path given one Data input. So for example if I wanted to know what ...