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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Kernel of the Laplacian on a compact manifold

Is there a way to characterise the kernel of the Laplace-Beltrami operator on a compact manifold without boundary? Or is it just "the set of functions $u$ such that $-\Delta u = 0$?"
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$(1,0)$-forms on a complex manifold

I'm reading in a paper: "Let $\theta$ be a $(1,0)$-form with respect to $I$ ($I \theta = -i \theta$)". Here $I$ is the almost complex structure. Any idea why it says $I \theta = -i \theta$ and not $I ...
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54 views

Integration of differential forms

I have just started to learn differential forms. Now, there is a concept of pulling integral back. I somewhat understood the procedure to do it. But, I don't understand why we do it and when to use ...
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Point on an ellipsoid closest to line

The $2D$ case is not a problem: $$\ P(t) =(x,y)= s + t v = <s_x+tv_x, s_y+tv_y> $$ $$\ F(x,y) = (\frac{x}{a})^2 +(\frac{y}{b})^2 -1 = 0 $$ $$ \nabla F(x,y).v =0 $$ Finally solve for $y$ in ...
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vote
1answer
49 views

Equation of a straight line in spherical coordinates

I'm trying to prove the angle sum formula for a triangle on the surface of a sphere. In order to do this I wanted to create a general triangle on the sphere, with one vertex at $\theta = 0$ and one ...
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2answers
36 views

Find the tangent and normal lines to the curve $\gamma(t)=(2\cos(t)-\cos(2t), 2\sin(t)-\sin(2t))$ at $t=\frac{\pi}{4}$

The normal line to a curve in the plane at a point $\mathbf p$ is the straight line passing through $\mathbf p$ perpendicular to the tangent line at $\mathbf p$. Find the tangent and normal lines to ...
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1answer
48 views

Prove that a straight line is the shortest curve between two points in $R^n$.

Let $p,q∈R^n$ and let $\gamma$ be a curve such that $\gamma(a) = p, \gamma(b) = q$, where $a$ < $b$. (a) Show that, if $\mathbf u$ is a unit vector, then $$\dot\gamma \cdot \mathbf u\leq ...
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21 views

Curve and Constant Curvature

I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant ...
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75 views

Why is it hold for types of operators?

We ‎let ‎the ‎state ‎space ‎be‎ ‏‎‎‎$ \mathcal{H} =‎ ‎‎H_{E}^{2}(0 , 1) \times L^2(0 , 1) $‎ equipped with the norm ‎ \begin{align} \| (f , g) \| = \int_{0}^{1} [ |f''(x)|^2 + |g(x)|^2] ‎\mathrm{d}x‎ ...
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How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...
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39 views

Equality in de Rham cohomology

Let $U_1,U_2,...,U_r$ be open sets in $\mathbb{R}^n$ such that $U_i\cap U_j =\emptyset$ for all $i \neq j$. Then prove, $H^k_{dR}(\bigcup_{i=1}^{r} U_i)=\bigoplus_{i=1}^{r} H^k_{dR} (U_i)$
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60 views

Shallow tent like soap film

A soap film circle in $x-y$ plane with center at origin can be carefully pricked with a blunt soapy pin at center and drawn out a little bit on $z$-axis forming a surface of revolution somewhat like a ...
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1answer
30 views

De Rham cohomology group

We know $m$-th de Rham cohomology group on $U$ is defined to be, $H^{m}_{dR}(U)=ker(d^m)/im(d^{m-1})$ where $d^m:\Omega^m(U)\to \Omega^{m+1}(U)$'s are usual exterior derivative maps. Now its saying ...
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2answers
66 views

is tangent bundle of $S^n$ an algebraic variety?

I have found somewhere that $T(S^n)$ is an algebraic variety in $C^n$. But now I can not recall the explicit form of this variety and the source of this information. It will be helpful if somebody ...
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2answers
39 views

Why is the irrational winding of the torus not locally path connected?

The irrational winding of the torus given by the map $f\colon\mathbb{R}\to T^2$ where $f(t)=(e^{it},e^{i\alpha t})$ for some irrational $\alpha$. Wikipedia mentions this is not a regular submanifold, ...
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80 views

Tangent bundle of $S^2$ not diffeomorphic to $S^2\times \mathbb{R}^2$

I am trying to show that the tangent bundle of $S^2$ not diffeomorphic to $S^2\times \mathbb{R}^2$. This is from an exam, where there is a hint stating that this is more than showing that $TS^2$ is ...
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18 views

Lefschetz number of a map and perturbations

I have a couple of doubts regarding Lefschetz numbers. I'm trying to answer the following related questions: Compute the Lefschetz number at $0$ of the map $f:\mathbb{C}\rightarrow \mathbb{C}$ ...
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2answers
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Is $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ diffeomorphic to $S^2$?

I was working on Problem 5-1 of Smooth Manifolds by Professor John Lee, and it lead me to wanting to show that $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ is diffeomorphic to $S^2$, and that is ...
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3answers
43 views

Length of Difference Curve

Let $\varphi : [a,b] \to \mathbb R^n$ be a curve, and for some partition $\pi = \{ t_0 = a, t_1, \ldots, t_m = b \}$ of $[a,b]$ set $$ l(\pi, \varphi) = \sum_{i=1}^m \| \varphi(t_i) - ...
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1answer
30 views

Orientation on $S^2$

I'd like to make sure I'm getting the proof of the following statement right: Let $S^2\subset \mathbb{R}^3$ be the unit sphere and define a vector field $N(x,y,z)=(x,y,z)$. Define a 2-form $\omega\in ...
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36 views

Smooth embeddings of the $2$-sphere

I have a past qual question here: given a smooth embedding $f \colon S^2 \to \mathbb{R}^3$, show that there must exist distinct points $p,q \in S^2$ such that the tangent planes to the embedded sphere ...
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Affine connection defined by a quotient manifold?

Suppose $G$ is a Lie group with affine connection $X,Y \mapsto\nabla_X Y\in C^{\infty}(G,TG)$, and $Q$ is a subgroup of $G$ such that $G/Q$ is also a nontrivial Lie group. Does this quotient manifold ...
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8 views

Fiber product of Fréchet manifolds

Let $X$, $Y$ be two manifolds modelled on Fréchet spaces. Let $f: X \longrightarrow M$, $g: Y \longrightarrow M$ be two smooth maps to a finite-dimensional manifold. Does the fiber product $X ...
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252 views

Convex surface on which any two points $a,b$ can be joined by a curve of length $(\pi/2-\epsilon)|a-b|$

I am trying to solve an exercise on page 13 of the book Metric structures on Riemannian and non-Riemannian spaces by Gromov. Construct a closed, convex surface $X$ in $\mathbb R^3$ such that any ...
2
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1answer
40 views

Constructing lagrangian submanifold of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. To keep it simple, let us take $M = \mathbb{R}^{2n}$ with linear coordinates $(x^1,\ldots,x^n,y^1,\ldots,y^n)$ and the standard symplectic form $\omega = ...
3
votes
1answer
26 views

Calculating the pullback of a $2$-form

I have a $2$-form given by $\omega = dx \wedge dp + dy \wedge dq$ and a map $i : (u,v) \mapsto (u,v,f_u,-f_v)$ for a general smooth map $f : (u,v) \mapsto f(u,v)$. I want to calculate the pullback of ...
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30 views

Constant curvature geodesic circles on a surface with constant Gauss curvature

Referring to: Curvature of geodesic circles on surface with constant curvature, Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE ...
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2answers
24 views

Inward and outward pointing tangent vectors?

If $M^n$ is a smooth manifold with boundary and $p\in\partial M$, then $T_pM$ is the disjoint union of inward and outward point vectors, and $T_p\partial M$. If $(U,(x^i))$ is a smooth boundary chart ...
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1answer
34 views

If a plane intersects a regular surface at exactly one point, then it is the tangent plane

Question Let a regular surface, $S$, intersect a plane, $P$, at only one point, $p_0 = (x_0, y_0, z_0)$ in $\mathbb{R}^3$. Show that the plane coincides with the tangent plane to the surface at ...
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votes
1answer
42 views

Degree and Lefschetz number of a function

I'd like to check if I got the following computation right. Let $f:RP^3\rightarrow RP^3$ be given by $[x_0:x_1:x_2:x_3] \mapsto [x_0^2:x_1^2:x_2^2:x_3^2]$ I would like to compute the degree and the ...
3
votes
1answer
30 views

Morse functions and indices of critical points

I'd like to check whether I got this exercise right. Let $S^1$ be the unit circle. Consider the function $f:S^1\times S^1\rightarrow \mathbb{R}$ given by $f(z,w)=Re(z)+Re(w)$. I'd like to check that ...
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0answers
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differential form: a question in Novikov's book

I met a problem when reading the book of Novikov et al. "Modern geometry, vol 1". In page 200 of the book (GTM 93, English version), they say for 2-form $F$, $$ (F\wedge F)_{0123}=-{1\over ...
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1answer
43 views

How to calculate scalar product of two gradients in indicial notation?

Does someone knows how calculate scalar product of two gradients and put the result in terms of nabla operator? . $(\vec\nabla{\gamma})\cdot(\vec\nabla{\gamma})$ = ?
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2answers
51 views

How does $v\Phi^1=\cdots=v\Phi^k=0$ imply $v\in\ker d\Phi_p$?

I'm confused about an immediate corollary in John Lee's Smooth Manifolds. Proposition 5.38 says Suppose $M$ is a smooth manifold and $S\subset M$ is an embedded submanifold. If $\Phi\colon U\to N$ ...
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1answer
41 views

Are the topology of a manifold and the topology induced by the metric of a manifold the same?

I am a physics student trying to understand in a rigorous way what a manifold is so please bear with me. Ok, so I am just learning what a topology is and from what I have understood up till now is ...
4
votes
1answer
39 views

connection laplacian on general vector bundles

As the title says, my question is about how to define the connection laplacian on general vector bundles. I think I understand how to define the connection laplacian on the tensorbundles: Let $M$ ...
3
votes
1answer
29 views

Geometric interpretation of $\beta_r(s) = \alpha(s) + r\mathbf{n}(s)$

Let $\alpha(s)$ be a smooth curve parameterised by arc-length and for fixed $r > 0$ define $\beta_r(s) = \alpha(s) + r\mathbf{n}(s)$, where $\mathbf{n}(s)$ is the unit normal vector to $\alpha$ at ...
3
votes
2answers
80 views

Symmetry group of the vector field $V=x \partial /\partial x + y \partial /\partial y$

I was trying to solve an exercise in one of Arnold's book that asks for the symmetry group of the vector field $V=x \partial /\partial x + y \partial /\partial y$, that is the diffeomorphisms $g$ of ...
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1answer
45 views

Unit circle can't be covered by one chart

I am hoping that someone can give me a proof showing why the unit circle cannot be covered by one coordinate chart, or a reference where I can find a proof.
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1answer
31 views

Covariant and contravariant bases on a diffeomorphism

If we allow two domains $\Omega, \bar{\Omega}\in \mathbb{R}^3$, allow $\mathbf{\Theta}: \Omega \to \mathbf{E}^3$ and $\mathbf{\bar \Theta}: \bar \Omega \to \mathbf{E}^3$ to be two ...
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Quaternion conversion

We have a normalized orthogonal co-ordinate frame travelling through the curve as in figure 1 below, from one end to other. Let us call starting end as A and ending end as B. What we know is initial ...
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Which part of differential geomety uses metrization theorems?

I learned three metrization theorems last year, which are Nagata-Smirnov,Smirnov and Bing. I thought these theorems are purely topological theorems, but i recently saw a post which says these ...
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1answer
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Why does $S^n$ satisfy the local $n$-slice condition? From Lee's Smooth Manifolds.

Example 5.9 on page 103 of Lee's Smooth Manifolds says the following: The intersection of $S^n$ with the open subset $\{x:x^i>0\}$ is the graph of the smooth function $$ ...
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0answers
34 views

A representation of a 1 form

Let $x, y, z$ be the usual coordinates on $\mathbb{R}^{3}$. Consider the 1-form on $\mathbb{R}^{3}$ given by $\phi = dx+ydz$. Do there exist smooth functions $u$ and $v$ such that $\phi=u\ dv$? Why? ...
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Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ I can approximate this to first order as: $$\tilde ...
4
votes
3answers
89 views

Interior product between differential forms and vector fields

I don't understand what is meant when someone writes that forms (or form fields) "eat" vectors (or vector fields). For example when I have a one form field ω=3dx+5dy+3xdz and a vector field ...
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35 views

Voisin's proof of Ehresmann's theorem

On p.221 of Voisin's book on Hodge theory, there are two claims: a) Let $B$ be a contractible smooth manifold. There exists a vector field $\chi$ on $B$ whose flow $\Phi_t$ is global and, given any ...
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1answer
32 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
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1answer
29 views

How to find the surface element for the cylinder $x^2 + y^2 = r^2$?

So if given a surface (cylindrical) which has radius r and equation $x^2 + y^2 = r^2$, I want to work out the line element for it. How do I get it? I know the final answer has to be $dS^2 = r^2dϕ^2 ...
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0answers
15 views

Mean curvature of even order

I read Antonio Ros, Compact Hypersurfaces with Constant Higher Order Mean Curvatures,1987. I don't understand following sentence from the second page 6th line. From the Gauss equation, we have ...