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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Proof of $d^\ast A =0$ where $D=d+A$ is Yang Mill connection

Recall $$ F =( dA_{ij} + A_{il}\wedge A_{lj} )\mu_i \otimes \mu_j^\ast $$ Hence if rank of $E$ is $2$, then $$ F= dA $$ since $A$ is skewsymmetric. If $D$ is Yang Mill connection then $ D^\ast F=0$. ...
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28 views

Some confusion about where vectors emanate from,

Take the plane x+y+z=0, for example. Then the vectors (0,0,0), (1,-1,0), (1,0,-1) "lie on this plane." And to find a normal to this plane, just compute the cross-product of any two independent ...
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113 views

What happens to small squares in Riemann mapping?

I have a square $S$, and I want to convert it to the unit disc $D$. The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion. ...
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1answer
45 views

Basis free way to show $\alpha(V,-)=\alpha(W,-)=0$ implies $\alpha([V,W],-)=0$?

Given a 2-form $\alpha$, I want to show that, if $X$ and $Y$ are vector fields such that $\alpha(X,Z)=\alpha(Y,Z)=0$ for all vector fields $Z$, then $\alpha([X,Y],Z)=0$ for all vector fields $Z$. ...
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45 views

About metric and the Ricci curvatrue

Recently, I met a question about the relation between $g$ and $-Ric_g$ on the Riemmannian manifold $(M^n, g)$. One said that "without loss of generality that by scaling $g$ in space we have $g \geq - ...
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1answer
6 views

Covariant derivative and tensor symmetries

Suppose we have a tensor field $T^{ab}$ such that $T^{ab} = T^{ba}$ everywhere. Then from the definition of the Riemannian covariant derivative in terms of a map between tensors, why must we then have ...
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31 views

The Product map of a Lie Group is a Submersion.

Problem 7.1 of Lee's Introduction to Smooth Manifolds (2nd Edition) reads: Show that for a Lie group $G$, the multiplication map $\mu:G\times G\to G$ is a submersion (Hint: Use Local Sections). ...
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2answers
31 views

Calculating the curvature of a tractrix

I'm trying to find the curvature of a tractrix expressed in the form $r(t)=(\sin{t},\cos{t}+\ln(\tan{(\frac{t}{2}))} $. From what I've found on the Internet it appears that people arrive at the ...
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1answer
35 views

Universal property of the tensor product

Assume $\Phi:V_1^*\times ...\times V_k^* \rightarrow L(V_1,...,V_k;\mathbb{R})$ is a multilinear map. $$\Phi (w^1,...,w^k)(v_1,...,v_k)=w^1(v_1)...w^k(v_k)$$ By the universal property of the tensor ...
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15 views

If $σ$ is an exact differential $1$-form on the plane, then the form $ω=σ+xdy$ is not exact

If $σ$ is an exact differential $1$-form on the plane, then prove that the form $ω=σ+xdy$ is not exact. In the previous part of the question we have calculated the integral of the differential ...
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1answer
30 views

Divergence of a tensor with respect to the Levi-Civita connection

In a Riemannian manifold $\mathcal{S}$ with metric $\boldsymbol{g}$, given a chart $\{x^a\}$, it is fairly easy to prove that the divergence of a vector field $\boldsymbol{w} : \mathcal{S} \to ...
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1answer
84 views

The derivative of a coordinate chart and outward orientation of vectors

I'm trying to get my head around the fact that $\phi$ is orientation preserving, due to $d\phi$, i.e. $d\phi$ sends outward vectors on $\partial \mathcal{M}$ to outward vectors on $\mathbb{H}^n$. ...
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1answer
47 views

Does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point?

As the question suggests, does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point? Many thanks in advance. To be clear, I'm using the statement of Brouwer's Fixed-Point ...
3
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1answer
41 views

An identity involving a Killing field

Does anyone know how to prove the following identity. We assume that $\Omega$ is a Killing field and $U, V$ are vector fields. Then $[\Omega ,\nabla _UV]-\nabla _U([\Omega, V])=\nabla _{[\Omega ...
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1answer
29 views

Given local smooth extensions, construct a global smooth extension

In Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1, he defines a function from a half-space $H^n$ to be $C^\infty$ if there is an extension to a neighborhood of $H^n$ that is ...
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1answer
37 views

Proving those are manifolds

Reading through McDuff, Salamon, I came across the following extract: We are supposing a Hamiltonian system has $n$ independent integrals $F_1,\dotsc,F_n$, that is there are $n$ functions $F_i$ ...
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55 views

If a line bundle admits a non-vanishing section then it is trivial

Suppose $\pi:E\to B$ is a line bundle. Let $s:B\to E$ a non-vanishing section, i.e. for every $b\in B$ $s(b)\ne 0$ and $\pi\circ s=Id_B$. I have to prove that the line bundle above is trivial. Idea: ...
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61 views

Non-integrability of distribution arising from 1-form and condition on 1-form

Suppose $M$ is a $(2k+1)$-dimensional manifold on which a 1-form $\alpha$ is defined. $M$ is termed as a contact manifold if the distribution arising from $\alpha$ is nowhere integrable, i.e. if: ...
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18 views

Classification of parallel spinors on the torus?

The $2$-torus $T^2$ has four different spin structures, typically parametrized by $\delta \in Z_2^2$. How many tuples $(\delta, g, \varphi)$ exist such that $\delta$ gives a spin structure on $T^2$, ...
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1answer
58 views

A fiber product is a fiber bundle

Let $F,B$ be topological spaces. A fiber bundle $E$ over the basis $B$ with fiber $F$ is a topological space $E$ endowed with a continuous surjection $\pi:E\to B$ such that there exists an open cover ...
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1answer
38 views

Diffeomorphism that pulls back the curvature tensor is an isometry?

I heard this statement somewhere. Can anyone provide a reference (or explanation of why this is true)? (I have also heard that the metric can be expanded as a power series in terms of the curvature ...
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55 views

Manifold, exist smooth nonnegative function with regular value at 0?

If $X$ is any manifold with boundary, then does there exist a smooth nonnegative function $f$ on $X$, with a regular value at $0$, such that $\partial X = f^{-1}(0)$?
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22 views

Dual Cone Construction $\{z \; | \;z \perp v \text{ for some } v \in \Lambda \}$

In a linear algebra computation, in order to estimate the second eigenvalue we consider a collection of vectors. Let $\Lambda$ be a cone in $\mathbb{R}^d$ then $$ \Lambda' = \Big\{z \;\Big| \;z ...
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1answer
25 views

Quotient group and kernel of canonical projection

Imagine we have a group $G$ acting properly and freely (as a group action $\Phi: G \times M \rightarrow M$) on a manifold $M$, then $M/G$ is a manifold and there is a smooth submersion $\pi: M ...
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52 views

Is the parametrization $(t^3,t^6)$, a reparametrization of $(t,t^2)$?

I have gathered that it is not. But I find it to obey all the conditions of being a reparametrization. Definition given in comments :P
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1answer
20 views

Proof: superharmonic function equal on $\partial D$ and at one point inside of D to its harmonic function, is harmonic on D (D compact)

I am looking for a proof (literature or short idea) for the following statement, which I have found in several sources: Let $M$ be a riemannian manifold, let $f:M\to\mathbb{R}$ be a superharmonic ...
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1answer
42 views

looking for the proof of a formula ($\mathbb{R}^3$ vector product) [closed]

does anyone know how to prove the following formula for vector product? $$(u\cdot\nabla)u=\nabla\frac{|u|^2}{2}-u\times(\nabla\times u).$$
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2answers
50 views

Is it possible to construct a smooth curve with fractional Hausdorff dimension?

It is known that fractal curves have fractional Hausdorff dimension. These curves are not smooth and have undefined length. However, is the converse true? If a curve has a fractional Hausdorff ...
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127 views

Geometric meaning of $\nabla_{[i}(x^i \nabla_{j]}x^j)$ and $(\nabla_{[i}x^i )\nabla_{j]}x^j$

While teaching myself tensor calculus I have come up with this proof of the 2-D hairy ball theorem. When trying to generalize this proof to higher dimensions I get the term $$\nabla_{[i}(x^i ...
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1answer
43 views

ricci tensor of 2-sphere $S^2$

Hi could someone show me explicitly how to compute the ricci tensor $g_{ij}$?
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2answers
47 views

Adjoint representation

I was just wondering why the adjoint representation of the Lie group $Ad$ and Lie algebra $ad$ are called representation. Maybe this word is derived from abstract algebra somehow, but I don't ...
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27 views

Question on the matrix of a Kaehler Metric in Normal Coordinates

I am currently studying normal coordinates on a Kaehler manifolds: Let $h$ be a Kaehler metric on a complex manifold $M$ and let $p \in M$. Let $(z_1,..,z_n)$ be a coordinate chart such that $h$ is a ...
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1answer
63 views

Number of smooth structures on $\mathbb{R}$ (not up to diffeomorphism)

On page 53 of Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1, Exercise 2-4 asks How many distinct $C^\infty$ structures are there on $\mathbb{R}$? (There is only one up ...
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2answers
41 views

Is Fermat's theorem about local extrema true for smooth manifolds?

Let $M$ be a smooth manifold and $f\colon M \rightarrow \mathbb{R}$ a smooth function. If $p\in M$ is a local extremum of $f$, does $p$ have to be a critical point?
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Maurer-Cartan Form on orthogonal group

I'm having trouble understanding an assignment regarding the Maurer-Cartan form on orthogonal matrices: Let $\text{O}(n)\subset\text{GL}(n,\mathbb{R})$ be the matrix Lie group of orthogonal ...
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1answer
13 views

Angle of tangent line and line $y=0,z=x$ is constant

Show that the tangent lines to the regular parameterized curve $\alpha(t)=(3t,2t^2,2t^3)$ make a constant angle with the line $y=0,z=x$. 1) The tangent line at each point is given, I believe, by ...
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1answer
188 views

What is the equation describing a three dimensional, 14 point Star?

I need to model a 14 point star. This is a three dimensional surface where there is a point at each of the eight corners of a cube and each of the six sides. The object is uniform (i.e. planar ...
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69 views

Can someone give me the spherical equation for a 26 point star?

This is the object that I am trying to find the volume of. This can be treated as a "26 point star". What I need is an equation to describe it. If anyone has that surface in spherical ...
2
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0answers
19 views

Local Lie derivative on $G$-space at the zero of a vector field

Let a Lie group $G$ act on a manifold $M$ and let $X\in Lie(G)$. For now suppose $G=T$ is a torus (but the answer to this question should hold for $G$ abelian). $L_X$ is a vector field on $M$, at a ...
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2answers
43 views

Compute the tangent space at the unit matrix

Compute the tangent space $T_pM$ of the unit matrix $p=I$ when $$(i)\,M=SO(n)\\ (ii)\,M=GL(n)\\ (iii)\,M=SL(n).$$ My attempt: I think I have computed the tangent space in the case that $M=SL(n)$. ...
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23 views

Difference between types of connections [closed]

For my background, I am familiar with the basics of differential geometry, especially Riemannian geometry, and in some more advanced topics relevant to physics, especially general relativity. Lately ...
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27 views

Imbedding a smooth manifold in Euclidean space - elementary proof for non-compact manifolds

The statement that there is an imbedding $$M \to \mathbb{R}^K$$ for some finite $K$ has a rather elementary proof provided that $M$ is compact. I have failed to find a proof of the same statement for ...
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28 views

standard unit tangent vectors to the unit sphere

I am reading the following I am having trouble in understanding how one can compute standard unit tangent vectors to $S^2$, i.e. the following Could anyone explain to me how $\theta$ and $\phi$ ...
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1answer
53 views

Am I right about this definition of submanifold?

Consider the following definition of submanifold: 1.5. $\ \bf Definition.\ $ A subset $M\subset\mathbf R^{n}$ is called a $\underline{\text{differentiable submanifold}}$ of $\mathbf R^n$ of ...
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1answer
20 views

smoothness, differentiability and continuity of a parametric curve

Hi I am reviewing basic vector calculus for geometry and stuck on two very elementary facts. First, A parametric curve $r=r(t)$, $a\le t\le b$ is called smooth if 1) $r'(t)$ exists; 2) $r'(t)$ is ...
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30 views

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

As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? I not a topologist by trade and I find the concept kind of hard to understand... Thanks ...
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2answers
160 views

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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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 ...