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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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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206 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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109 views

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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48 views

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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97 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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117 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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321 views

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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111 views

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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191 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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57 views

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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147 views

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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49 views

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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95 views

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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189 views

De Rham cohomology of $\mathbb R^3$ without lines and a circumference

I am trying to calculate De Rham cohomology of the following spaces: $X=\mathbb R^3\setminus r$ where $r$ is a line; $Y=\mathbb R^3\setminus (r \cup \gamma)$ where $r$ is a line and $\gamma$ is a ...
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90 views

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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77 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 ...
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187 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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89 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 ...
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156 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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263 views

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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224 views

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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85 views

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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116 views

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 ...
2
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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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134 views

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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84 views

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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81 views

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 ...
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67 views

How to prove an isotopy relative to a point exist?

Let $M$ $ $ be a differential manifold, and $f$ a diffeomorphism on $M$ which is isotopic to $id$. Assuming that $x\in M$ is a fixed point of $f$ and the orbit of $x$ under the isotopy is a trivial ...
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338 views

Constant rank map

Suppose $F:M\to N$, $M,N$ smooth manifolds and $M$ connected. I proved that for each $p$ in $M$ there exists smooth charts near $p$ and $F(p)$ in which the coordinate rapresentation of $F$ is linear. ...
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312 views

Connections and curvature on general fiber bundles

Let $\pi: E \to M$ be a fiber bundle with fiber $F$. We can define a connection to be a projection $v: TE \to VE$ satisfying $v \circ v= \text{Id}$, where $TE$ is the tangent bundle of $E$ and $VE$ is ...
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79 views

About Thom theorem (representation submanifold for $H_{n-2}(M)$)

Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold. And in the Harper and ...
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49 views

List of minimal surfaces embedded in the 3-sphere

Is the set of possible areas of closed, embedded, minimal surfaces of the 3-sphere discrete?
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822 views

Top deRham cohomology group of a compact orientable manifold is 1-dimensional

Let $M$ be a compact smooth orientable manifold of dimension $n$. I am looking for a simple proof that $H_{dR}^n(M) \cong \mathbb R$. Equivalently, an $n$-form which integrates to 0 is exact. I can ...
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316 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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61 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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95 views

Complex structure on the product of two complex Kähler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kähler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
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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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119 views

Twisted tori: discrete and continuous

Taking the advice of Mariano Suárez-Alvarez, I moved this question from MO to MSE: Motivation Let me introduce twisted (discrete) tori: Consider the Cartesian graph product $\mathcal{C}_n = C_n ...
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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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68 views

How to apply Gauss's theorem when the metric is unknown

Let $f:U \to \mathbb{R}^3$ be a surface, where $U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$ Consider the two closed square regions $F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, ...
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159 views

Hopf theorem for non-orientable manifold

I have an exercise which says that :extend the Hopf-Poincare theorem for non-orientable manifold with the indication using the double covering. I have got stuck for long time, so I don't know if ...
2
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190 views

A question about a definition of Ricci curvature

Let me first quote the definition of Ricci curvature from Wikipedia. Let $M$ be an n-dimensional Riemannian manifold equipped with its Levi-Civita connection $\nabla$. The Riemannian curvature tensor ...
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59 views

Reconstructing paths on the sphere from the ratio of acceleration to velocity

Given a path $\gamma:[0,1]\to \mathbb C$, we can determine $\gamma$ from information about its derivatives. For example, $\gamma$ is determined by $\gamma(0), \gamma'(0)$, and $\gamma''(t)$. This ...
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178 views

Conjugate function reverse angle in a neighborhood

Can you please help me with this question. Prove that if conjugate of $F$ is analytic in a neighborhood of $z_0$ in $C$ then $F$ reverses angles at $z_0$. Can anyone please explain that for me? what ...
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92 views

laplacian for functions problem with the integral on manifolds

I'm following the proof of the local expression for the Laplacian on a compact manifold and I'm having problems understanding how the integral on a manifold translates into an integral in $R^n$, in ...
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194 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 ...