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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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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70 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 ...
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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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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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0answers
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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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 $\phi$ ...
2
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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 $M=SL(n)$. ...
2
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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
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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2answers
59 views

Mean Curvature flow: Evolution equation of any invariant symmetric homogeneous polynomial with input the Weingarten map.

I have the following evolution equations realted to mean curavture flow, with the induced metric $g=\{g_{ij}\}$, measure $d\mu$ and second fundamental form $A=\{h_{ij}\}$: 1)$\frac{\partial}{\partial ...
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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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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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2answers
162 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.
4
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0answers
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 ...
6
votes
4answers
110 views

Show that Lie bracket is in Lie algebra?

Let $so(3)$ be the Lie Algebra of $SO(3)$ and $R\in SO(3); \Omega_1,\Omega_2 \in so(3)$ and $\Omega_n = \frac{d}{dt}R_n(t)$ at the point $t=0$. So $\Omega_n$ is the tangent vector of the curve ...
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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 ...
3
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1answer
42 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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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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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 ...
3
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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 ...
4
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0answers
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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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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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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2answers
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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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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0answers
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
votes
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 ...
6
votes
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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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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0answers
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 ...
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
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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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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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 ...
5
votes
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 ...
3
votes
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
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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0answers
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 ...
5
votes
1answer
139 views

This theory proof about instability of a point of equilibrium is not understandable for me, any help?

-This theory is irritating me, because I don't understand it's logic. Theorem: If in some neighboorhood $\mathbb O (0)$, exists a continuous, differentiable function $V(X), V(0)=0,$ such that the ...
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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 ...
2
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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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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 $P$. ...
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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 ...
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votes
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 ...
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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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 ...
2
votes
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. ...