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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Holonomy group as a quotient group of $\text{GL}(k,\mathbb{R})$

Currently, I'm reading the script of a Global Analysis lecture at my university. There, we look at a real vector-bundle $(E,\pi,M)$ of rank $k$ with a given connection $\nabla$ and define the ...
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36 views

How to create a simple closed curve homotopic to the trefoil knot?

How to create a simple closed curve that is homotopic to the trefoil knot $\overrightarrow{\alpha} (t)= \left ( \left (3+ \cos (3t) \right) \cos (2t),\left (3+ \cos (3t) \right) \sin (2t),\sin (3t) ...
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120 views

De Rham cohomology of the pointed plane

i try to work out some examples for de Rham cohomology, but i have some problems: I want to figure out what $H^k(\mathbb{R}^2\setminus\{0\})$ is and want to generalize this to arbitrary finite points ...
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38 views

Confusion about canonical basisvectors of the tangentspace of a manifold

Let $M$ be a $n$-dimensional manifold. We want to deduce a basis of $T_xM$ for $x\in M$. for $x\in M$ we can find an open neighbourhood $U$ such that $x\in U$ and an homeomorphism onto an open set ...
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207 views

Canonical orientation of a complex manifold

It is clear to me that a complex manifold is orientable. However is there a canonical choice of one of the two possible orientation or is it just a matter of convention? And, if it is a convention, is ...
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170 views

Geodesics of this metric

I have to calculate the geodesics of the metric: $$\left(\matrix {1 &0\\0& x^2 }\right)$$ I've been able to derive its equations, which are: $$\ddot x -x\dot y ^2=0$$ $$\ddot ...
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1answer
74 views

Geodesic radius of curvature

I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula. $\frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G}$ where $s$ is the ...
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1answer
77 views

Is this a tangent bundle and what is the meaning of this exercise

I intend to solve the following exercise but I would like to have some help with understanding the ''big picture'': Exercise. Describe a natural 1 to 1 correspondence between elements of $SO(3)$ ...
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274 views

Why is the constant relating the chern class and curvature form always $2\pi i$?

I'm reading Milnor's book on Characteristic Classes. In Appendix C, Milnor shows the invariant polynomial of the curvature form and the Chern class differ by powers of $2\pi i$. He first shows that ...
3
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1answer
112 views

Coincidence about nabla?

I was surprised to notice that gradient of function and Levi-Civita connection have the same notation, i.e. nabla sign $\nabla$. Moreover, extending any connection on tensors, one let it be ...
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1answer
109 views

Alexandrov embedded disc

We say that a compact surface $\Sigma$ is Alexandrov embedded via an immersion $f:\Sigma \rightarrow \mathbb{R}^3$ if there exist, $X$ a 3 manifold and an immersion $F: X \rightarrow \mathbb{R}^3$ ...
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78 views

Help understanding a proof in differential geometry

I was reading John Milnor's Topology from the Differentiable Viewpoint and there's a proof of the fundamental theorem of algebra at the end of the first chapter that I don't fully understand. I can ...
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51 views

What is $T^*(\mathbb{A}^1)$?

Let $\mathbb{A}^1$ be the affine line and $T^*(\mathbb{A}^1)$ the contangent space of $\mathbb{A}^1$. What is $T^*(\mathbb{A}^1)$? Is $T^*(\mathbb{A}^1) = \mathbb{A}^2$? Thank you very much.
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63 views

Question about a notation. Norm of the derivative of a function at a point

Given is an analytic function from $M$ to $N$, both equipped with conformal Riemannian metric, say $g$ and $h$ resp. What might the $h$ norm of the derivative of the function at a point mean? ...
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2answers
254 views

Are all metric tensors diagonal?

If I understand correctly, one way to get the components of a metric tensor (treating it like a matrix here) is to look at the ds interval. Isn't that interval always in terms of sums of dr squared ...
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1answer
163 views

Geometric interpretation of Laplace's formula for determinants

Coming from the geometric point of view, the determinant of an $n \times n$-Matrix computes the volume of an parallelepiped spanned by the columns of the matrix. In context of this question, let the ...
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2answers
204 views

Interpretation of $p$-forms

Let $M$ be a smooth manifold, let $C^{\infty}(M)$ be set of all smooth functions from $M$ to $\mathbb R$ and let $Vec(M)$ denote the set of all vector fields on $M$. A $1$-form on $M$ is a ...
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1answer
66 views

Topological subspace in $(S^{1})^{n}$

Studying the set of solutions of a particular linear system associated to a matroid, I notice that is it possibile to determine the topology of the quotient and identify it as a subtorus of ...
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1answer
85 views

finding a basis of tensor product of dual space

Let $U$ and $V$ are two vector spaces and $U^*$ and $V^*$ are respective dual spaces . Let {$e^i:1\le i \le n$} and {$a^j:1\le i \le n$} are bases for $U^*$ and $V^*$ respectively . Then author says ...
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1answer
186 views

Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
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2answers
46 views

Solving $\left\{\begin{matrix}u'v''-u''v'=0 \\ R^2u'u''+v'v''=0 \end{matrix}\right.$.

Given that $u,v$ are functions of $t$, $R$ constant, solve $\left\{\begin{matrix}u'v''-u''v'=0 \\ R^2u'u''+v'v''=0 \end{matrix}\right.$. When trying to find geodesic on cylinder, I get this ...
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125 views

Avoiding vertical vectors in tangent spaces.

Let $i: [0,1]\hookrightarrow\mathbb{R}^3$ be a smooth embedding. Can I find arbitrary small perturbations $j$ of $i$ s.t. $j(0)=i(0)$, $j(1)=i(1)$ and $j'(t)$ is never a multiple of $(0,0,1)$? More ...
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49 views

Any elementary derivation of the Pfaff integrability condition?

Suppose in $\mathbf{R}^N$ we have a one-form field, $ \theta = \sum_{i=1}^N \theta_i d x_i $. The Pfaff integrability condition is $d \theta \wedge \theta = 0$. Is it possible to give an ...
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1answer
92 views

Curvature of a manifold

If we need to find the curvature of a manifold, does that mean we need to find a sectional curvature or a holomorphic sectional curvature in the case of a complex manifold?
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1answer
37 views

What line does ω project vectors onto?

I have just started learn differential form from the bachman book (page 29)and I found some difficulties in the following problem in 2nd part. Let $ω(<dx,dy>) = −dx + 4dy$. 1. Compute $ω(<1, ...
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105 views

Question about line of curvature

If $\alpha$ is a planar geodesic on surface $M$, show that $\alpha$ is a line of curvature. My try: $\alpha$ planar imply torsion=0, and binomial vector is constant. Since ...
3
votes
1answer
223 views

Orientation double cover

Let $M$ be a manifold and let $\bigwedge^\text{top}TM$ be the top exterior product of the tangent bundle. Then this becomes a line bundle. Let $g$ be any metric on $\bigwedge^\text{top}TM$ and define ...
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1answer
87 views

Question about Tangent vectors and coordinate change on manifolds

We can describe a vector on a manifold $M$ of dimension $n$ as follows: Let $p:I\rightarrow M$ with $I$ open interval in $\mathbb{R}$ be a curve in $M$. Now look to $p_0=p(0)$. Locally we can find a ...
2
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1answer
72 views

Spherical-Coordinate Reference Frame

my problem looks apparently easy but I can figure out a solution. I'm going through a paper about the Boltzmann equation and I got stuck with this change of coordinates. The original formula for Q ...
3
votes
1answer
145 views

Smooth Manifold, covered by 2 Charts is orientable if the Intersection is Connected

I came across this Question: Atlas on a smooth manifold that contains 2 charts in which Professor Lee commented that this Proposition is true only if the Intersection of the two Maps is connected, so ...
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1answer
80 views

What can be said about functions of constant Hessian determinant?

Let $f:\mathbb{R}^2\to \mathbb{R}$ with $\det \nabla^2f = 1.$ Let's also assume that $\nabla^2 f$ is positive-definite (which we can do WLOG by adjusting the sign of $f$). What can we say about $f$? ...
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1answer
71 views

A question about the condition of Frobenius theorem

I puzzled about the condition of Frobenius theorem: Condition FR1: Let $X$ be a manifold, $E$ is a subbundle of $TX$,vector fields $ ξ,η $ lie in $E$(i.e. $ ξ(x),η(x)\in E_x $),then bracket $[ξ,η]$ ...
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2answers
46 views

How to prove a $k$-$1$ differential form is simple

I've been both trying to prove and looking for a proof in a couple of book and on the Internet, and I can't find it. How can I prove that a $k$-$1$ differential form defined on a $k$ dimensional ...
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2answers
116 views

Idempotent operators over the exterior algebra

I am wondering if there exists a (reasonably) well-known set of operators $A_i$ over the exterior algebra such that $\{A_i,A_j\} = \frac{1}{2}(A_i +A_j)$, where $\{X,Y\}=(XY+YX)/2$.
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Splitting of the tangent bundle of a vector bundle and connections

Let $\pi:E\to M$ be a smooth vector bundle. Then we have the following exact sequence of vector bundles over $E$: $$ 0\to VE\xrightarrow{} TE\xrightarrow{\mathrm{d}\pi}\pi^*TM\to 0 $$ Here $VE$ is ...
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2answers
238 views

Understanding of the Regularity Condition in the definition of regular surfaces

I am having some difficulty in understanding the meaning/motivation of the regularity condition in the definition of regular surfaces. The definition (restricted to $\mathbb{R}^2$ and ...
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2answers
311 views

Application of Christoffel symbol in differential geometry

When self-studying differential geometry, I find my book involves some clumsy, troublesome calculation about Christoffel symbol when proving theorem, which in fact doesn't have the symbols. I wonder ...
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1answer
52 views

Strainers in Alexandrov spaces

I am reading the section on Strainers in Burago, Burago and Ivanov's book "A Course in Metric Geometry". I have been struggling with the proofs of some of the lemmas. On Lemma 10. 8. 13, the authors ...
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2answers
84 views

The property of an integral manifold of differential form

Let's define the $1$-form $\eta$ on $\mathbb{R}^3$ by formula: $$\eta = A(x,y,z)\;\mathrm{d}x + B(x,y,z)\;\mathrm{d}y + \mathrm{d}z \text{.}$$ And let's assume that $$\mathrm{d}\eta \wedge \eta = ...
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1answer
221 views

Proof behind $S^n\cong SO(n+1)/SO(n)$

I have been trying to understand the fact that $S^n \cong SO(n+1)/SO(n)$. I believe I have the intuition correct at this point; consider the case when $n=2$ as we have $S^2 \cong SO(3)/SO(2)$.: We ...
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1answer
55 views

Applying Green's theorem for a line integral of a vector field

Integrate the vector field $F(x,y)=(e^y+\frac{1}{y+3},xe^y-\frac{x+1}{(y+3)^2})$ over a curve that goes from $(-1,0)$ to $(-1,2)$ to $(0,1)$ to $(1,2)$ (in a linear fashion). Now, I'm almost certain ...
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102 views

De Rahm Cohomology of Complex Grassmannian

Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$ ...
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1answer
60 views

Visualizing the level sets for this function

Let $F: T^{2} \to \mathbb{R}$ be given by $(x_{1}, x_{2}, x_{3}) \mapsto x_{2}$. Recall that $$T^{2} = \{(x_{1}, x_{2}, x_{3}) : \left(\sqrt{x_{1}^{2} + x_{2}^{2}} - R \right)^{2} + x_{3}^{2} = ...
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41 views

Terminologies for induced connections

Given a Riemann manifold with a Kozul/Affine connection, if you take any subbundle of the tangent bundle there is an induced connection given by applying the ambient Kozul connection and projecting to ...
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1answer
169 views

Connected sum in an ambient space

Let $M$ be a smooth c connected $m$-manifold, $N_1$, $N_2$ two smooth disjoint connected $n$-submanifolds with boundary, both contained in the interior of $M$ (if necessary). Assume that $M\setminus ...
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48 views

Homology of smooth loop spaces of spheres

I'm looking for a reference on the homology of $C^\infty(S^1,S^n)$. So far I've only been able to find references for spaces of maps which are $C^k$ or in a Sobolev class. I also have references which ...
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47 views

Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
3
votes
1answer
186 views

Why $S^1\times S^{2m-1}$ carries a complex structure.

Let $S^n$ denotes $n$-sphere, then why $S^1\times S^{2m-1}$ carries a complex structure.
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Dimensions of the cohomology groups of certain complicated space

Let contruct the space $X$. We take the complex projective space $\mathbb{C}P^2$, pick two points $p_1, p_2 \in \mathbb{C}P^2$ and remove two small, disjoint, open $4$-balls $B_j$ centered at $p_j$. ...
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1answer
374 views

Principle G bundles v.s. Flat G connection

What is the difference between Principle G bundles v.s. Flat G connection? I heard that for a discrete group $G$ (in physics, or a finite group $G$ in math), the principle G bundles is the same ...