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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What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration?

As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? Many thanks in advance.
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1answer
43 views

Question about Normal Coordinates on a Kaehler Manifold

I am currently reading FY Zheng's textbook, "Complex Differential Geometry". In section 7.4 Proposition 7.14, he is trying to prove thata metric $h$ Kaehler is equivalent to the statement, "For any $p ...
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0answers
29 views

Easy examples of non-arithmetic lattices

I'm starting to look a bit more at discrete subgroups of Lie groups, particularly lattices. A lot is written about arithmetic lattices of Lie groups, and examples abound. It appears that much less is ...
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48 views

Homogeneous Isotropic Riemannian Manifolds

In John Lee's book Riemannian Manifolds on page 33, Lee writes "Clearly a homogeneous Riemannian manifold that is isotropic at one point is isotropic at every point". It seems that he means that he ...
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1answer
91 views

Any use of advanced Abstract Algebra in Differential Geometry?

I believe that if someone is going to continue their studies and doing research on Differential Geometry's topics, would never need advanced Abstract Algebra (or maybe not even undergraduate level of ...
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1answer
63 views

How to express the second fundamental form in terms of deformation second gradient

Suppose we have a surface $\Omega$ with prescribed principal curvatures, $\kappa_1$, $\kappa_2$, say. An isometric deformation ${\bf r}:\Omega\rightarrow\mathbb{R}^3$ maps the surface into ...
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1answer
70 views

First variation of an action?

I'm working on a problem and I must compute the first variation of an action. Let $\Omega$ is a 2-form on a semi-Riemannian manifold $M$ and $f$ is a smooth function and $\Gamma$ is an 1-form on $M$. ...
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1answer
67 views

Accounting for signs in divergence thm. on Lorentzian manifold

I am trying to learn about integration in Lorentzian manifolds (I will use signature -+++) and have some problems. Oft quoted (in books for GR) form of divergence theorem is: $\int _U div( X ...
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1answer
30 views

Differential geometry: $|\alpha(t)|=c\ne 0 \iff \langle \alpha(t),\alpha'(t)\rangle = 0,\forall t\in I$

Let $\alpha: I\to \Bbb R^3$ be a regular para curve. I want to prove that: $|\alpha(t)|=c\ne 0 \iff \langle \alpha(t),\alpha'(t)\rangle = 0,\forall t\in I$ Now $|\alpha(t)|=c\ne 0$ means that this ...
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38 views

Solving non-linear equations in a chosen subspace

I'm trying to find the root $\mathbf{f(x)=0}$ to the following sets of equations $$ f_1(x,y,z) = x^\prime - \frac{x}{\sqrt{x^2+y^2+z^2}} = 0 \\ f_2(x,y,z) = y^\prime - \frac{y}{\sqrt{x^2+y^2+z^2}} = ...
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1answer
24 views

examples: q is a conjugate point of p, but geodesic $pq$ is unique

Let $(M,g)$ be a Riemannian manifold, $p\in M$. It's well known that if the geodesic connecting $p$ to $q$ is not extendable at $q$, then either the geodesic connecting $p$ to $q$ is not unique, or ...
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21 views

Existence of real structure on CY m-fold

Suppose $M$ is Calabi-Yau $m$-fold with complex structure $J$, Kahler form $\omega$, metric $g$ and holomorphic $m$-form $\Omega$. What are the conditions on $M$ for the existence of a map $\sigma: M ...
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1answer
70 views

Proving that the tangent vector of a simple closed curve rotates by $ 2 \pi$

I am trying to prove that if $\gamma(t)=(x(t),y(t))$ ,a function from the closed interval $[0,1]$ to $\mathbb{R^2}$ is a simple closed unit speed curve such that $\gamma '(0)=\gamma '(1)$. Then the ...
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1answer
49 views

The integral of a closed form along a closed curve is proportional to its winding number

Source: Guillemin-Pollack Exercise 4.8.2. Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 - \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 - \{0\}$. Prove that$$\oint_\gamma ...
4
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1answer
65 views

Modifying a smooth function with respect to conditions on its partial derivates

Let $\{U_i\}_{i\in I}$ be a locally finite collection of open subsets of $\mathbb{R}^n$, $K_i\subseteq U_i$ compact subsets, $\epsilon_i>0$ positive real numbers and a nonnegative natural number ...
4
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1answer
59 views

is the pullback operator associated to a flow bounded in L^2?

Let $M$ be a smooth compact manifold with a finite Borel measure $m$. Let $\{f_t\}_{t\in\mathbb R}$ be a $C^1$ flow on $M$. That is, a $C^1$ function $$ \mathbb R\times M\ni(t,x)\mapsto f_t(x)\in M $$ ...
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1answer
68 views

Are closed simple curves with that property necessarily circles?

This is a more interesting follow-up to the question Are closed simple curves with this property necessarily circles? Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple $C^1$ convex curve and ...
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0answers
33 views

Looking for a reference that explains connections and curvature by double tangent space

I'm looking for a book or a set of lecture notes on differential manifolds that explain connections (Levi-Cevita connection, prinicipal connections) and curvature on an abstract manifold from the ...
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55 views

Problems understanding this proof

This is an extract from Duistermaat's Fourier integral operators. I'm having a hard time understanding the proof. My questions are three: How do I use the implicit function theorem to ...
2
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1answer
23 views

Pulling back a connection to a curve

What does it mean to pull a connection back to a curve? For example, if I take the connection $\nabla s = ds$ on the trivial bundle $\mathbb R^2 \times \mathbb R^2$ over $\mathbb R^2$, and the curve ...
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0answers
25 views

Computing the differential of multiplication by $M$ on $U(n)/O(n)$

Suppose $M\in U(n)$. Then multiplication by $M$ induces a smooth action on $U(n)/O(n)$. How can we compute the differential of this map? If $M$ were acting on a matrix group, then of course the ...
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1answer
87 views

Holonomy computation in $S^2$

If $\gamma$ is a closed Loop in $S^2$ and $p\in S^2$, where $\gamma$ is the boundary curve of some region $X$ in $S^2$ (and $\gamma$ satisfied some regularity conditions), someone told me that the ...
2
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1answer
48 views

Two different descriptions of a complex torus

I came across the following description of a (1-dimensional) complex torus while learning about Calabi-Eckmann manifolds: For a fixed $\alpha \in \mathbb{C} \setminus \mathbb{R}$, the subgroup $Z = ...
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1answer
115 views

A problem from Arnold's book

I was reading Arnold's ODE book, there is written as corollary that Consider the differential equation $\frac{dx}{dt}= v(t, x)$ with $t\in \mathbb{R}$ and $x \in\mathbb{R^n}$. Then there exists a ...
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1answer
42 views

Curvatures in differential geometry-interpretation

The are various notions of curvatures in differential geometry: soft such as full curvature tensor for a given connection (which is tensor of type $(1,3)$), Ricci curvature tensor (type $(0,2)$ ...
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42 views

Differential forms defined by integration

Let $\omega_1,\omega_2$ be two n-forms on a $n$-dimensional manifold $M$. Now, imagine we have for every open $N \subset M$ that $$\int_{N}\omega_1 = \int_N \omega_2.$$ Can anybody show me how to ...
2
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1answer
75 views

Poincaré hyperbolic geodesics in half-plane and disc models

EDIT1: The derivation of geodesics of the two models follows in a straightforward manner from the metric. For the half-plane we have in Cartesian coordinates $$ ds^2 = (dx^2 + dy^2)/y^2 \tag{1} ...
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1answer
44 views

The explicit expression for integral of forms

Could anyone please help me with the following three questions? They are simple questions, but I am confused. With a $2$-form $F=\frac{1}{2}F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$ in 4 dimension, what is ...
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0answers
58 views

Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot $ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
6
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1answer
558 views

De Rham cohomology for non-compact manifolds

Let $M$ be a non-compact differential manifold. It is true that in general $H^q_c(M) \neq H^q(M)$, where $H^q_c$ is the de Rham's cohomolgy with compact support group and $H^q$ is the usual de Rham's ...
3
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0answers
163 views

In differential geometry, is there established notation for the stuff that $\mathbb{R}^n$ is equipped with?

Given $v,p \in \mathbb{R}^n$, I will write $T_p(v)$ for $v$ viewed as a vector based at $p$. So: $$ T_p(v) \in T_p\mathbb{R}^n$$ Question 0. Is there an accepted notation for what I'm denoting ...
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2answers
19 views

Derivative and vector for start of curve are orthogonal to fixed vector, hence the curve is

Let $\alpha:I\to\Bbb R^3$ be a parameterized curve and let $v\in \Bbb R^3$ be a fixed vector. Assume that $\alpha'(t)$ is orthogonal to $v$ for all $t\in I$, and that $\alpha(0)$ is also orthogonal ...
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1answer
24 views

Construction of partitions of unity in Warner

On p. 11 of Warner's Foundations of Differntiable Manifolds and Lie Groups, he discusses partitions of unity. The theorem says Let $M$ be a differentiable manifold and $\{U_\alpha: \alpha \in A ...
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1answer
32 views

Proving this result on tangent spaces to foliations

Reading through Lee's introduction to smooth manifold, I bumped into this result: I've tried to prove it, but have gotten stuck. A foliation is basically slicing $M$ into $k$-dimensional ...
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1answer
108 views

Two nonevident implications in a proof

I am reading part of Lee's introduction to mainfolds. I got to the following proposition. I am having trouble between the two displayed lines of the proof. Precisely, my questions are: How does ...
3
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0answers
55 views

Quick question: Determinant bundle is Cartier?

$X$ is an algebraic surface (i.e. compact complex, which embeds in a projective space). $V$ is a vector bundle of rank 2 over $X$. Why is $\det{V}=\mathcal{O}_X(D)$ for some divisor $D$? Is there a ...
3
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1answer
70 views

I am confused by the different definitions of manifolds.

I'm currently learning manifold from Do Carmo's Riemannian Geometry. This is his definition of differentiable manifold: But this is different from what I saw in wiki: A differentiable manifold ...
3
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1answer
68 views

Integrating the Hopf invariant for $\pi:S^3\to S^2$

I've been working on the last part of problem 9., chapter 9 in Nakahara's Geometry, Topology and Physics all day, with no success, and am in need of some assistance. We are asked to compute the Hopf ...
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1answer
54 views

Why can't that be an uncountable union?

I'm reading part of Lee's Introduction to manifolds. I have come to the following proposition. The proof then continues, and I will read the rest shortly. I was just wondering: why can't ...
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1answer
84 views

Lie Derivative of a section on a vector bundle

I'm still trying to figure out how to do the Lie derivative of a Jacobian. (c.f. earlier unanswered post). If I know how how to do Lie derivatives on section of vector bundles, that would be ...
2
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1answer
41 views

If $\nabla_1$ and $\nabla_2$ are Levi-Civita connections for a metric on the smooth sphere, then their curvature tensor would recover the radius…?

I am a little confused by an idea suggested to me: putting a connection on a sphere doesn't specify a metric geometry - it remembers notions like straightness of paths, but not length of paths. Let ...
3
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1answer
33 views

Geodesics in upper half-plane model of $\mathbb{H}$

On this page in Schlag's book on complex analysis, he is discussing the upper half-plane model of $\mathbb{H}^2$. He says for all $z_0\in \mathbb{H}$ $$\{T'(z_0) \mid T \in PSL(2, \mathbb{R}) ...
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1answer
27 views

Holonomy computation in a sphere

Let $S^1$ be the unit sphere in $\mathbb R^3$, and let $$C=\{(r\cos t, r\sin t, h)\colon t\in \mathbb R\}$$ with $r^2+h^2=1$ be a circle in $S^2$. I want to compute the holonomy around this circle. ...
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0answers
46 views

Integration by parts on manifold with a boundary

Suppose $C$ is a 3-form, and $G$ is a 4-form defined by $G = dC$. Also, $M_{11}$ is an 11-dimensional manifold (without a boundary), $W_{6}$ is a 6-dimensional submanifold of $M_{11}$ and ...
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2answers
69 views

Given two points in a manifold, can i find compact path-connected set that contains both

Suppose we are given two points in path-connected smooth manifold. My hypothesis is that we can find path-connected compact set that contains both. I have no idea how to prove it, in fact I don't know ...
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0answers
35 views

Reference for theorems in Hirsch

In Hirsch's differential topology, we find the following theorems on page 31: 4.1 Theorem: Let $M$ be a $C^r$ $\partial$-manifold and $N$ a $C^r$ manifold, $r \geq 1$. Let $f : M \to N$ be a $C^r$ ...
2
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2answers
36 views

How to find cartesian coordinate of velocity of particle on the trajectory, $Ax^2 +2Bxy +Cy^2=1, A,B,C >0.$

Consider a particle with constant speed $|w|=w_o$ moving on trajectory $Ax^2 +2Bxy +Cy^2=1, A,B,C >0.$ Could anyone advise me how to express cartesian coordinates of $v$ in terms of $x$ and $y \ ?$ ...
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1answer
42 views

Invariance of form under flow

Take a two-dimensional symplectic submanifold $M$ in $\mathbb{R}^3$. Now, I want to show that the symplectic form $\omega$ is invariant under the Hamiltonian flow $\phi^{t}: M \rightarrow M.$ Is it ...
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41 views

The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ bilinear … so is it tensor like?

The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ linear in both components... so is it a tensor of some kind? I know (I think) it is not a ...
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0answers
14 views

Applications of normal coordinates?

I am looking for applications of normal coordinates (from the exponential map) on a Riemannian manifold, since I am trying to familiarize myself with this notion. Presently the only one that I know ...