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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Commutator of two “special” conformal Killing fields

Theorem 1.7 of Schottenloher's "Mathematical Intro to CFT" says that every conformal Killing field on a connected open subset $M$ of $\mathbb{R}^{p,q}$ for $n=p+q>2$ is of the form $$X^{\mu}(x) = 2 ...
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On differential geometry in Hilbert spaces

Suppose that $H$ is a Hilbert space and $M\subset H$ is a closed subset with non-empty interior and smooth boundary, whatever smooth boundary could mean. I wonder if the normal vector is onto on the ...
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253 views

Manifolds: A definition of the Gradient an Algebraic Tangent vector over charts, how to show equivalence?

Let X be an n-dimensional differentiable manifold and $p \in X$ . Let $(U, h, V )$ for X around p with coordinates $(x_1 , . . . , x_n )$ in V , and let $v_i , i = 1, . . . , n$ , be the basis of ...
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Chern class of tautological line bundle

I'm studying characteristic classes from the Chern-Weil construction (via connection and curvature). I'm trying to compute some simple examples. Let $E$ be the tautological line bundle over projective ...
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An orbit of a group action and the implicit function theorem

Suppose that a Lie group $G$ acts smoothly on a manifold $X$. We can easily prove the following theorem by using the constant rank theorem, which is a stronger theorem than the implicit function ...
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Fundamental Group, Piecewise Smooth Curves, Consevative Fields

Let M be compact Riemannian manifold. X be a vector field on M. I believe that work done by X any piecewise closed curve is zero, iff, the same is zero for a particular finite set of loops. I believe ...
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Following a polyline along the surface of a polygon that is twisted

I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ...
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64 views

Stoke's theorem application to curl theorem. I did. Please can you check it?

Now, I need to apply stoke's theorem to curl theorem. My teacher gave a hint. Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$ $dim(M)=2$ M is the subset of $\Bbb ...
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208 views

Show that the projection map is Orientation preserving iff n is even

My question is that Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere $U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ...
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51 views

Trivialization of a path of tamed almost complex structures

I am wondering if the following result is true: Let $(V,\omega)$ be a symplectic vector space and $\{J_t\}_{0\leq t\leq 1}$ a smooth path of almost complex structures on $V$ which are tamed by ...
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82 views

Pole of differential

Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$. We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as ...
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Curvature form projective spaces

Let $T\mathbb{C}P^n$ tangent bundle over $\mathbb{C}P^n$. We have an hermitian metric on $T \mathbb{C}P^n$ defined as $h=\frac{dzd\bar{z}}{(1+|z|^2)^2}$. If we consider Levi-Civita connection we can ...
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371 views

Chern classes tangent bundle.

I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over $\mathbb{C}P^n$ ($T\mathbb{C}P^n$) and develop the theory beginning from this ...
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124 views

Cotangent bundle of a complex projective space

How does the cotangent bundle of a complex projective space looks like? Is that an Einstein manifold?
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126 views

“Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
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How to define local sections on associated bundles?

On a $G$-principal bundle $P$ it's easy since $G$ has a preferred element, namely $e$ (identity), but there is no such element in arbitrary suitable space $F$ to construct local sections for the ...
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117 views

characterization of the differentiable functions over a regular surface

Let $S$ be a regular surface. And let $f:S\to \mathbb R$ be a differentiable function It's not hard to prove that if $ W$ is an open set of $\mathbb R^3$ such $ V\subset S\subset W$, and $f:W\mathbb ...
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Problem solution hint about boundary of boundary of chains from Arnold' book mathematical method

On his book Mathematical Methods of Classical Mechanics, (Chapter 7, Section 35, Problem 10), Arnold asks to show that the boundary of boundary of any chain is zero. He gives hint saying: by the ...
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275 views

Torsion of a closed curve

Why the total torsion of a closed curve on a sphere is zero? What is the meaning of total torsion? Is it mean that we should calculate the torsion from point A to point A (i.e. since the curve is ...
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82 views

Does the derivative of the derivative depend on a choice of connection?

Let $X,Y$ be smooth manifolds and consider the infinite-dimensional manifold $$ C^\infty(X,Y) $$ of smooths maps $f: X \to Y$. Note that there is an infinite-dimensional vector bundle $E$ over this ...
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116 views

Reason for defining Riemannian curvature tensor and torsion tensor in particular way

I saw how Riemannian curvature tensor and torsion tensor are defined, but I am not sure why they are defined that way. In 3-dimensional euclidean space with ordinary multivariable calculus, the ...
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205 views

A question on tangent plane (from Do Carmo)

From 'Do carmo Differential Geometry of curves and surfaces' On page 89, #9. Show that the parametrized surface S given by $$ \text{x}(u,v)=(v\cos{u},v\sin{u},au) $$ Compute its normal vector ...
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Divergence and curl united?

In my post, In 2D we can define $$div(F) = \lim_{C\rightarrow 0}\frac{1}{A}\int_C F \cdot \hat{r} dC$$ $$curl_z(F) = \lim_{C\rightarrow 0}\frac{1}{A}\int_C F \cdot \hat{v} dC$$ Where $C$ is a ...
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How can we calculate the basis for right invariant vector fields from basis left invariant vectr fields

I want to calculate the right invariant vector fields from left invariant vector fields. The fact I am using is that for a driftless system $\dot X= XA$ we have $\dot Y= -AY$ where $Y=inverse Y$
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174 views

How to calculate the Gaussian curvature of a non-embedded surface

I know how to calculate the Gaussian curvature of an embedded surface using first and second fundamental forms, but how does one calculate the curvature of a non-embedded surface like the hyperbolic ...
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95 views

smoothness structure on a set

I'm reading Milnor's 'characteristic classes', and in the first chapter he defines smoothness structure on a set M, which is confusing for me, as what follows:(these are not Milnor's words so please ...
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72 views

an example of a curve such that…

Given differentiable functions $k(s),\tau(s)$ with domain $s\in (a,b)$, there exist a differentiable function $\gamma:(a,b)\to \mathbb R^3$ such that $k,\tau$ are it's respectively curvature and ...
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115 views

A question from Hamilton's Ricci Flow book by bennett chow

On page 3 of the book before exercise 1.2, has written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free condition results ...
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61 views

deRham cohomology of a manifold with covering space $S^{n}$

Let $\pi: S^{n}\rightarrow M$, $n>1$ be a covering map, $M$ being an orientable manifold. Show that $H^{k}_{deR}(M)=0$ for $1\leq k<n$. I know how to do for $H^{1}_{deR}$, but my argument fails ...
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do Carmo: near isolated zeros, killing field tangent to geodesic spheres

Exercise 3.5b of do Carmo's Riemannian Geometry asks the reader to prove that given a Killing field $X$ on a manifold $M$, an isolated zero $p$ of $X$, and a normal neighborhood $U$ of $p$ in which ...
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Schwarz Lemma in Differential Form

Suppose $w=f(z)$ is a conformal self map of $\mathbb{D}$. From Schwarz Pick Lemma we have $|\frac{dw}{dz}|=\frac{1-|w|^2}{1-|z|^2}$. Could any one explain me In differential form how this becomes ...
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187 views

Circle tangent bundle over $S^{2}$

Let $S_{r}^{2}$ be a sphere of radius $r$ and let $TS_{r}^{2}$ be its tangent bundle. If $SS^{2} = \{(x,p)\in TS_{r}^{2}| |p|=1 \}$ be the circle tangent bundle of non zero radius . Then are there ...
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Dual connections, bracket

If $\nabla$ is a torsionfree connection and $(\nabla_{X}J)Y=(\nabla_{Y}J)X$, J- an almost complex structure, and $\nabla_{X}^{*}Y:=J\nabla_{X}(JY)$ its dual connecion. Is it correct to conclude that ...
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prove that is not conformal map

I have fun with this website. It is very helpful and great. I have a question in geometry. I try to solve it but it needs imagination to figure out the point on the surface. This question makes me ...
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Why is that quantity a constant?

Help needed! What have I done wrong here? Given the metric $$ds^2 = dr^2+r^2d\theta^2$$ And $$R_{ij}=\nabla_i \nabla_j f(r)$$ where $\nabla_i$ is a covariant derivative, and $f=f(r)$ is a scalar ...
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Integral over a Funnel in Fermi coordinates

Suppose we are in the Hyperbolic plane, defined as $$ \{w = u + iv \mid v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,. $$ I am given a funnel $F$. This object is isometric to a ...
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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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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 ...
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184 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 $\omega\in\Omega^2(S^2)$: $$\int_{S^2}\omega=\int_{B^2}F_+^*\omega-\int_{B_2}F_-^*\omega$$ ...
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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 ...
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152 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' ...
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Use of Implicit Function Theorem to provide examples of Manifolds

Suppose we know Definition 1 and Theorem 1, and want to prove Theorem 2 given below. Definition 1: A subset $M$ of $R^n$ is called an k-dimensional manifold (in $R^n$) if for each point $x\in M$ the ...
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Correct use of substitution rule for Integration on Riemannian manifolds

Let $(N,g_N)$ be a Riemannian manifold and let $\psi: M \rightarrow N$ be a a diffeomorphism. Now I know how the Riemannian metric on $M$ defined by the pull-back of the metric on $N$ looks like (this ...
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Are there more types of differential in mathematics?

I am familiar with two types of differential normal differential: $$d^2x^a$$ covariant differential: $${\mathcal D^2x^a}=d^2x^a+\Gamma^a_{bc}dx^bdx^c$$ (where the covariant differential is broken ...
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69 views

Non-Linear Beltrami Vector fields

Consider two concentric toruses, and let $\Sigma$ be the domain interior to the greater torus and exterior to the smaller torus. Is it possible to find a vector field $\mathbf{b}$ satisfying the ...
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Can 24 lines on a cubic surface be realized as 24 identical spiral rods?

It's possible to put 24 lines on a cubic surface. 27 lines is possible, but I don't have a great picture for that surface. It turns out that the 24 lines can be built with Zome. I'm thinking that ...
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69 views

Convex polyhedral decomposition of spheres

Is there a decomposition of $S^2$ into $k$ (geodesically) convex polyhedra that are congruent to each other? What about $S^n$ for $n>1$? Remarks: A polyhedron is defined as an area enclosed by a ...
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Mean curvature operator in S^n

Consider the sphere $S^n$. By using the stereographic projection we can identify $S^n \setminus N$ with $\mathbb{R}^n$, where $N$ is the North pole of $S^n$. The metric then is given by ...
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Convexity of a function

Suppose we have $F: R^n \longrightarrow R$ , $P: R^n \longrightarrow R^n$ and $G: R^n \longrightarrow R$ all nice- let's say given by polynomial and $P$ is invertible - such that $F(x) =G( P(x) )$. ...
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Pushforward/derivative of map between surfaces in $\mathbb{R}^n$

If $f:M \to N$ is a smooth map between compact closed hypersurfaces $M$, $N \subset \mathbb{R}^n$, does it make sense to write the pushforward as $Df$, the total derivative? Because usually we require ...