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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Showing diffeomorphism between $S^1 \subset \mathbb{R}^2$ and $\mathbb{RP}^1$

I am trying to construct a diffeomorphism between $S^1 = \{x^2 + y^2 = 1; x,y \in \mathbb{R}\}$ with subspace topology and $\mathbb{R P}^1 = \{[x,y]: x,y \in \mathbb{R}; x \vee y \not = 0 \}$ with ...
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119 views

Visualizing diffeomorphisms

This is probably a really basic question (hence my asking it here as opposed to MO). In a comment to a question on mathoverflow ...
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85 views

frontier of class $C^{1}$.

Studying the Divergence Theorem (Gauss theorem), found the definition of frontier of class $C^{1}$. Which means? That is, the one which is a set with boundary of class $C^{1}$? Can give reference ...
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198 views

Approximation of Lipschitz functions on Riemannian manifolds

Let $ (M,g) $ be a Riemannian manifold ($ g$ Riemannian metric) and let $ f: M \rightarrow R $ be a Lipschitz function (with respect to $ g $) with compact support. I want to study if it is possible ...
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1answer
210 views

Understand the Hyperbolic space

I've been trying to find the expression for the metric of the hyperbolic n-space, $\mathbb H^n$. For $n=2$ I've found (e.g. here) that $$ds^2=\frac{dx^2+dy^2}{y^2}.$$ But for $n>2$ I can't seem to ...
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44 views

$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...
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1answer
425 views

Intuitive meaning of immersion and submersion

What is immersion and submersion at the intuitive level.what can be visually done in each case?
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1answer
143 views

Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
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99 views

Existence Theorem for Geodesics

The text I am reading now defined geodesics to be those curves that satisfy the following differential equation: $\ddot{\gamma}^k(t)+\dot{\gamma}^i(t)\dot{\gamma}^j(t)\Gamma^k_{ij}(\gamma(t)) = 0$ ...
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679 views

Where can I learn about complex differential forms?

So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd ...
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1answer
65 views

Show that $\kappa_{\delta} = \frac{\kappa}{1 - r\kappa}$

$\delta_r(t) = \gamma(t) + r U(t)$ is a parallel curve to a parametric curve $\gamma(t): I \rightarrow \mathbb{R}^2$ at distance $r$. I have already shown that $\delta_r' = (1 - r \kappa) \cdot | ...
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140 views

Riemannian metric. Help with notation.

I was just reading about the hyperbolic space (upper-half plane model) and i'm getting kind of confused about the notation for the Riemannian metric. The half-plane is defined as $$ H = \{(x,y) \in ...
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196 views

Structure of a $ C^{\infty} $-manifold

I was studying differentiable manifolds (an introduction) and found the following example, but I am confused. Example The function \begin{align} f: &\mathbb{R}^{3} \to \mathbb{R}, \\ f: ...
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1answer
132 views

Rademacher theorem for Riemannian manifold

Let $M$ be an open set of $\mathbb R^n $ and let $ ds^2 $ be some metric on $M$. Let $ d $ be the distance induced by $ ds^2 $ on $M$. If $ f $ is a Lipschitz function with respect to $ d $, is it ...
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1k views

Gentle introduction to fibre bundles and gauge connections

To better understand papers like this for example, which makes heavy use of fibre bundles and gauge connections to represent gauge fields, I am looking for a nice introduction to this topic. The only ...
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1answer
67 views

Nadirashvili surface (part 3)

The article that I'm considering is 'Notes sur la démonstration de N. Nadirashvili des conjectures de Hadamard et Calabi-Yau' by Pascal Collin and Harold Rosenberg. In the proof of the appendix (of ...
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1answer
95 views

Special Kähler manifolds

If we have complex vector space $V=T^{*}C^{m}$ with standard complex symplectic form $\Omega =\sum_{i=1}^{m}dz^{i}\wedge dw^{i}$, and if $\tau : V\to V$ is standard real structure of $V$ with set of ...
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1answer
687 views

Showing that left invariant vector fields commute with right invariant vector fields

I'm trying to prove that if $G$ is a Lie group, $X$ is a left-invariant vector field on $G$, and $Y$ is a right-invariant vector field on $G$, then $[X,Y] =0$. When I imagine what it means to be ...
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1answer
206 views

elementary questions about differential forms

QUESTION 1: So I know that if $\omega$ is an alternating $p$-form for odd $p$ on some vector space $V$, then $\omega\wedge\omega = 0$. But...isn't the same true for any $p$? Ie, take for example $p ...
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2answers
161 views

Restriction of the Laplace Beltrami operator

Given the expression of the Laplace-Beltrami operator $\Delta M$ on a Riemannian manifold $M$ , is there any method for determining the expression of the Laplace-Beltrami operator $\Delta N$ where ...
5
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1answer
268 views

Restriction of a differential form to an isotropic submanifold

From Analysis and Algebra on Differentiable Manifolds, first edition, exercise 2.6.4., question 1 (slightly edited for this post): Let $\vartheta$ be the canonical 1-form on the cotangent bundle $T^* ...
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The standard connection in a regular surface is symmetric

The concrete setting: Let $M\subset \mathbb{R}^3$ be a regular surface. A vector field $X$ in $M$ is a differentiable function $X: M\to \mathbb{R}^3$ such that $X(p)\in T_pM$. Here we are taking the ...
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1answer
99 views

Complex projective manifolds and holomorphic mappings

Let $X\subset\mathbb{P}^2$ be a complex manifold defined by a homogeneous polynomial of degree $d>3$. Let $$\phi:\mathbb{P}^1\rightarrow X$$ be a holomorphic map. Show that $\phi$ is constant. ...
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837 views

How are infinite-dimensional manifolds most commonly treated?

At a summer school I recently attended, infinite-dimensional manifolds popped up. I have never worked with them before (although I'm very familiar with finite-dimensional manifolds). The lecturer at ...
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41 views

Nadirashvili surface (part 2)

The article is 'Hadamard and Calabi Yau conjectures on negatively curved an minimal surfaces' Nadirashvili. In the proof of proposition 4.3 it asserts that the function y is holomorphic. I'm not sure ...
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363 views

Are the Ricci and Scalar curvatures the only “interesting” tensors coming from the Riemannian curvature tensor?

In studying Riemannian geometry, one learns about the Riemann curvature tensor $$Rm(X,Y,Z,W) = \langle \nabla_X\nabla_YZ -\nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z, W\rangle$$ and its various symmetries. ...
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1answer
141 views

What are all isometry classes of the 2-sphere?

In topology, one learns how to classify the compact surfaces up to homeomorphism. And in fact, since "homeomorphic" and "diffeomorphic" coincide in dimension 2, we can classify the compact (smooth) ...
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1answer
57 views

Diffeomorphisms to $S^n$

Is $S^4$ diffeomorfhic to $S^2\times S^2$? Moreover. Is $S^n$ diffeomorphic to some cross product of manifolds $X\times Y$ for $n\geq2$? Is there a elemental topological invariant to let me see ...
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2k views

Roadmap to study Atiyah-Singer index theorem

I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
3
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1answer
154 views

What is the norm of the gradient of $f$ in normal coordinate?

Let $M$ be a Riemannian manifold and $f$ a smooth function on $M$. The Bochner formula proved in Schoen-Yau's book "lectures in Differential Geometry": (prop. 2.2) $$ \Delta |\nabla f|^2(p)=2\sum ...
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1answer
100 views

Nadirashvili surface

I'm referring to the article of N. Nadirashvili "Hadamard's and Calabi-Yau conjectures on negatively curved and minimal surfaces". In the proof of proposition 4.3 author use a theorem of Walsh. Now ...
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1answer
103 views

Why this equality must holds for minimal surfaces?

When minimizing a surface area with respect to a fixed volume $V$, I found in some notes that the parametrization $X: U \longrightarrow \mathbb{R}^3$ must satisfy the equality $\iint_U (2H - \lambda) ...
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1answer
676 views

where can I find solutions to A comprehensive introduction to differential geometry by Spivak?

I have tried google and I fail to find solutions to the exercises in the book A comprehensive Introduction to differenial geometry volume I by Spivak. Does anyone know about a site with solutions to ...
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Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
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1answer
170 views

Compute the differential of a smooth map

Let $S\subseteq \mathbb{R}^3$ be an oriented regular surface and let $N$ be a field of normal unitary vector on $S$. We consider the map $F:S\times \mathbb{R}\rightarrow \mathbb{R}^3$ defined by ...
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3answers
443 views

How to prove that the set is not a regular surface?

I know that $$S=\{(x,y,z)\in \mathbb R^3: z^2=x^2+y^2\}$$ is not a regular surface, bacause it has a vertex in $(0,0,0)$. But how to show it precisely? Maybe here is usefull the theorem that a ...
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how to cover a manifold almost everywhere?

I pose the following question: Given a connected (maybe we need compactness) manifold or regular surface, can we find a single parametrization $\chi:B\to M$ from the open ball to the manifold, such ...
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1answer
227 views

Surface Parameterizations

I've been reading Manfredo Do Carmo's Differential Geometry of Curves and Surfaces and was wondering what are the conditions that need to hold for a surface parameterization as this is not defined in ...
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42 views

Approximation/Representation of local stable manifolds

I will give two preceding theorems and the question, which uses both, follows afterwards: Let $M$ be a smooth compact Riemannian manifold of dimension $n$ with a smooth measure $\mu$. $T_{x}M = ...
3
votes
2answers
99 views

Composite of an immersion with the inverse map of another immersion is a diffeomorphism

Let $U\subset \mathbb{R}^k$ be an open set, $n>k$ and $\varphi_1,\varphi_2 : U\to \mathbb{R}^n$ be immersions, meaning continuously differentiable such that the differential taken in any point of ...
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2answers
169 views

set of all singular points of a map

I was interested to know whether set of singular points of a smooth map forms a manifold? for example if $f:M\rightarrow\mathbb{R}^2$ is a smooth map. and I am trying to find the singular points and ...
2
votes
2answers
281 views

Gentle introduction to quasi-geodesics

Compared to the concept of geodesics the concept of quasi-geodesics seems to be substantially harder to grasp and digest. I was given a promising hint to the concept of quasi-geodesics here but the ...
4
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1answer
385 views

Symplectic 2-Sphere

Consider the sphere $S^2\subset\mathbb{R}^3$ in cylindrical coordinates $(\theta, z)$ (away from poles $z=\pm 1$) with symplectic structure $\omega=d\theta\wedge dz$. I want to show that the vector ...
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145 views

Do the polynomial germs generate all the ring of germs?

I'm trying to understand some equality that comes up in stability theory involving sets of germs and I think I need a result like the next one, so if anyone knows anything about this and helps me it ...
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1answer
141 views

Is every compact hypersurface contained in a sphere which it touches twice?

Let $M\subset \mathbb{R}^{n+1}$ be a compact $n$-manifold. There exists, then, a smallest $n$-sphere containing $M$, and it must touch it in one point. Must it touch it twice? This seems quite ...
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80 views

Exact sequence involving the nabla operator

Recently I noticed that $$0 \longrightarrow \Bbb R \overset{\text{const.}}\longrightarrow \mathcal{C}^\infty(\Bbb R^3,\Bbb R) \overset{\text{grad}}\longrightarrow \mathcal{C}^\infty(\Bbb R^3,\Bbb R^3) ...
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1answer
161 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
2
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1answer
58 views

Bounding the injectivity radius from below.

Let $(M, g)$ be a finite-dimensional Riemannian manifold, and let $S \subseteq M$ be a compact set. I want to prove the following fact: There exists a number $\epsilon > 0$ such that the ...
3
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2answers
87 views

Critical paths for length cannot have kinks.

This problem is in Spivak's Differential Geometry (Ch.9 #37), and he gives a sketch of a proof which I have been unable to finish. So let's compute $\frac{dL(\overline{\alpha}(u))}{du}\mid_{u=0}$ ...
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322 views

Geodesics on a polyhedron

Which sequences of adjacent edges of a polyhedron could be considered to be a geodesic? The edges of a face most surely will not, but the "equator" of the octahedron eventually will. But for what ...