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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105 views

What is 'target manifold'?

I saw in a lecture about O(3) sigma model something about 'target manifold', but I do not know what is it. Is there any book I could learn about that?
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64 views

Way distinguishing whether or not complex manifold

$SU(3)$ has dimension 8. Why is this not a complex manifold ? Thank you in advance.
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68 views

Trajectory of circumference [circle] rolling down any given curve

How should i go about describing mathematically the path traced by the center of a circumference [circle] rolling down (or up) any given curve described by $y = f(x)$? The solution for a linear ...
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1answer
151 views

showing that a local diffeomorphism is a local isometry using first fundamental form

In differential geometry, there is a theorem about 1st fundamental form : A local diffeomorphism $f:S_1 \rightarrow S_2$ is a local isometry $\Leftrightarrow$ For any patch $\sigma$ of $S_1$, ...
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1answer
41 views

Meaning of CR-Automorphism

What is the meaning of the CR-Automorphism and CR-Manifold? I tried to find the definition from the web. Is it Continuous Real ....? Thanks.
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0answers
90 views

Problem about finding Gaussian curvature

The problem is about Gaussian Curvature When we define $M=r(u,v)$ be the surface parametrized by $(u,v)$ in $\mathbb{R}^3$ and $N$ be a unit normal vector of $M$. In my lecture we denote Gaussian ...
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1answer
77 views

curves and constant length

Find all curves in $\mathbb{R}^2$ having the following property: the segment of the normal straight line between curve and the x axis has constant length. If $\alpha (t)=(x(t),y(t))$, I found ...
3
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1answer
85 views

Finding all alternating bilinear $T$ that preserve a certain group of isometries of $\mathbb{R}^{n+1}$

Let $$G=\left\{\begin{pmatrix} H & 0 \\ 0 & 1\end{pmatrix} \ | \ H\in O(n), HJ=JH \right\}\subset \mathrm{Lin}(\mathbb{R}^{n+1},\mathbb{R}^{n+1}) $$ where: $n=2m$, $J$ is the standard complex ...
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81 views

Does the boundary of a handle decomposition obtain a handle decomposition?

Let $M$ be the $4$-manifold $D^4\cup2\text{-handle}\cup\ldots\cup2\text{-handle}$, where the attachment of the handles is specfied by an oriented framed link $L=L_1\cup\ldots\cup L_n\subseteq S^3$. By ...
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1answer
360 views

Question in Do Carmo 1-2

In Manfredo Do Carmo's Differential Geometry of Curves and Surfaces, Section 1-2, he asks: Let $\alpha: I \to \mathbb{R}^3$ be a smooth curve that does not pass through the origin.If $\alpha(t_0) ...
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2answers
206 views

How can I compute the area of a geodesic triangle?

How can I compute the area of a geodesic triangle in a Riemannian 2-manifold? If the Gauss curvature $K$ is constant and positive I can take the Gauss-Bonnet theorem to obtain ...
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1answer
102 views

Confused by local isometries

I think I am a little confused about the notion of a local isometry of Riemannian manifolds. Let's say I have a manifold $(M,g)$ where $g$ is the Riemannian metric. Take a chart $x:U \rightarrow ...
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1answer
123 views

Transport theory basics: can't understand solid angles

I don't understand something in transport theory: $$P(x,\vec{w})=p(x,\vec{w}) \cos\theta \, dw \, dA$$ This is the number of particles flowing across a differential surface element in the direction ...
2
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1answer
343 views

solid angle of polyhedron

I have an interesting thought when I draw polygon and 3D polyhedron. My question is: Can I know the number of Face, Edge, and Vertex from a given 'space angle constraint'? For example, a vertex a cube ...
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1answer
50 views

Req. for Definition:Twisting Number of Curve in Contact Structure

All: I'm reading a paper that makes mention of the twisting $tw (\gamma,S) $ , where $\gamma$ is a simple, closed Legendrian curve in a surface $S$ , and $S$ is embedded in a contact 3-manifold ...
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1answer
36 views

submersion, density-perserving,

In Rerference http://www.jstor.org/stable/2243135, Theorem 1 says Let $t: M \rightarrow N$ be a $C^\infty$ function. Then $t$ is density-preserving if and only if $t$ is an almost submersion. I want ...
2
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1answer
117 views

Flowlines of blobs

I have the following formula for blobs/metaballs, which is said to be the same as the one used for electromagnetism: \begin{gather} f(x,y,z) = \frac{d(A,B)}{\sqrt{(x-xA)^2 + (y-yA)^2 + (z-zA)^2}} \\ ...
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1answer
220 views

How to compute Bochner laplacian $\Delta=\nabla^*\nabla=\sum \nabla_{e_i}$?

I'm struggling with proving that Bochner laplacian can be described by the following formula similar to the standard laplacian formula from calculus: $$\Delta = \sum_i \nabla_i^2,$$ where $\nabla_i = ...
3
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1answer
103 views

Non commutative tensor products / simple example?

In O'Neil's "Semi-Riemannian Geometry" it is stated that if $A$ is a $(r,0)$-tensor field and $B$ is $(0,s)$-tensor field then it is ""from the definition'' that $A \otimes B = B \otimes A$. I still ...
5
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1answer
109 views

Mixed Partials from Peter Petersen's book

I am trying to understand how mixed partials are defined for a function $\gamma : \mathbb R^m \rightarrow M$, where $M$ is an $n$ dimensional manifold, from Peter Petersen's "Riemannian Geometry" ...
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1answer
62 views

Conditions of existence for an annulus' embedding with special properties

Well, I have this issue and I'll be very thankful to any advices, hints or recommendations. Suppose we have a domain $D$ in $\mathbb{R}^3$ which is described by inequalities: $$ x^2 + y^2 \geqslant ...
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243 views

Is $ds$ a differential form?

I am somewhat confused as to whether $ds$ (line element) is actually a differential form... we have (in $\mathbb{R}^2$): $$ds^2 = dx^2 + dy^2$$ Differential 1-forms are supposed to be linear ...
3
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1answer
56 views

The smooth deformation

$M$ is a connected smooth manifold and $p$ is a fixed point on $M$. For a null-homotopic smooth loop $\gamma$ at $p$, can we find a smooth deformation, that is, a smooth function $f :[0,1] \times ...
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98 views

On an open set $U ⊂ \mathbb{R}^n$ show that the exterior derivative d is the only operator $d : Ω^p(U) → Ω^{p+1}(U)$ satisfying

On an open set $U ⊂ \mathbb{R}^n$ show that the exterior derivative $d$ is the only operator $d : Ω^p(U) → Ω^{p+1}(U)$ satisfying: $d(ω + η) = dω + dη$; $ω ∈ Ω^p(U), η ∈ Ω^q(U) ⇒ d(ω ∧ η) = dω ∧ η + ...
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1answer
129 views

Can a tangent vector extend to a vector field?

Let $M$ be a smooth manifold and $p\in M$. I would like to know whether any tangent vector $X_p \in T_pM$ extends to a vector field over $M$. If so is it unique? How can I construct it? Thank you.
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1answer
142 views

O'Neil's problem 2.10 on Tensor Derivations. Literature on the subject.

I am having real trouble trying to understand a problem in O'Neil's "Semi-Riemannian Geometry" and I can't find much literature on the subject. I will expose the problem and I will be grateful to ...
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2answers
97 views

Parametrized curve in $\mathbb{R}^n$

Let $\mu: [x,y]\to \mathbb{R}^n$ be a parametrized curve in $\mathbb{R}^n$ such that $\mu(x)=x_1$ and $\mu(y) = y_1$. Show that for any constant vector $v$ where $\lVert v \rVert=1$ then: ...
16
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1answer
313 views

Legendre Transformation of a Lagrangian in Classical Mechanics

I have some questions about the Legendre Transformation of a Lagrangian in Classical Mechanics to the Hamiltonian: We start with a Lagrangian $L(q,\dot{q})=\frac{\langle \dot{q} , \dot{q}\rangle }{2} ...
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1answer
53 views

Geometry finding area problem

A regular 2N -sided polygon of perimeter L has its vertices lying on a circle. Find the radius of the circle and the area of the polygon.
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302 views

Restriction of smooth functions.

Consider the following question: Suppose that $X$ is a subset of $\mathbb{R}^n$ and $Z$ is a subset of $X$. Show that the restriction to $Z$ of any smooth map on $X$ is a smooth map on $Z$. (Note: A ...
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1answer
42 views

Light transport theory - can't visualize basics

I don't understand why the following should hold in light transport theory: let p(x) be the number of particles per unit volume at the point x, then the total number of particles P(x) in a small ...
2
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1answer
58 views

Cartan subalgebra of compact group as “annihilator” of a single element

Let $G$ be a compact Lie group. The Cartan subalgebra $\mathfrak{h}$ can be defined to be a maximal abelian subalgebra of the Lie algebra $\mathfrak{g}$ of $G$. I know this is not the standard ...
3
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2answers
594 views

A curve parametrized by arc length

Let $C$ be a plane curve parametrized by arc length by $\alpha(s)$, $T(s)$ (unit tangent vector) and $N(s)$ (unit normal vector). Prove that $$\frac{d}{ds} N(s)=-\kappa(s)T(s).$$ I know that ...
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1answer
122 views

curvature of objects on a 2-d digital image

For a 2-d digital grey-scale image, I assume that any object, such as lines, edges, contours, is just 2-d stuff, isn't it? Or since each pixel of any object has a grey-scale value, which makes them ...
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1answer
148 views

Tensor of type $(1,1)$

I am a bit stuck here with this problem: If $A$ is a tensor of type $(1,1)$ and has the same component with respect to every basis, show that $A$ is a multiple of $\langle\,\,,\,\,\rangle$. I tried ...
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4answers
237 views

Formula for the curvature of a curve

Find a formula (provide your answer in terms of $f$ and its derivatives) for the curvature of a curve in $\mathbb{R}^3$ given by $\{(x,y,z)\ | \ x=y, f(x)=z\}$. How will I be able to do this ...
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1answer
43 views

Reference request for Euclidean metric in hyperspherical coordinates

As per the title, I'm just looking for a reference with a convenient derivation (or at minimum, description) of the Euclidean metric in hyperspherical coordinates. The specific cases of polar or ...
3
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2answers
207 views

Curvature of a regular parametrization

Prove that if $\mu: [a,b] \to \mathbb{R}^n$ is a regular parametrization of a curve then the curvature at $\mu(t)$ is given by: $$\kappa(t) = ...
6
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1answer
197 views

Problem 4-25 from Spivak's Calculus on Manifolds

I am reading through Spivak's Calculus on Manifolds and have come across a technicality in one of the problems that is annoying me. It is Problem 4-25, the statement of which is Let $c$ be a ...
5
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1answer
176 views

Vector calculus and Frenet-Serret equations

I have shown the first two equality and I am working on the showing the 1st equals the 3rd. \begin{alignat*}{4} \frac{1}{\rho}\hat{\mathbf{{n}}} &= \frac{d\hat{\mathbf{{u}}}}{ds} &{}= ...
6
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1answer
171 views

Geometric Interpretation: Parallel forms are harmonic

Let $(M,g)$ be a Riemannian manifold. The canonical volume form $\mu=\sqrt{\det g_{ij}}\mathrm{d}x^1\wedge\dots\wedge\mathrm{d}x^m$ is parallel w.r.t. the induced Levi-Civita conection $\nabla$ ...
2
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1answer
238 views

Prove that $S$ is a sphere.

Let $S\subset {\mathbb{R}}^3$ with the following properties: $1.$ For any line $l$: $|l\cap S|=2$ or $|l\cap S|=1$ or $|l\cap S|=0$ $2.$ For any plane $P$ $P\cap S=\text{circle}$ or $|P\cap S|=1$ ...
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4answers
788 views

How to convince a high school student that differentials don't work like fractions in general?

It all started when I tried to convince a 10th grader that if $f$ is a function defined on $\mathbb{R}^n$ the differential is defined by: $\large \displaystyle df = ...
7
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1answer
284 views

Turning higher spheres inside out

I know that one can turn a sphere $S^2$ in $\mathbb{R}^3$ inside-out having at each time an immersion of $S^2$ into $\mathbb{R}^3$. It is called Smale paradox. There is beautiful animation about that. ...
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0answers
96 views

Stratification of a smooth map

I am trying to do an exercise. Namely, find the Thom-Boardman stratification of the smooth map $f(x,y,a,b,c,d)=x^2y+y^3+a(x^2+y^2)+bx+cy$, where $a,b,c$ are parameters. As I have seen, this is also ...
3
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0answers
199 views

A probable inspiring proof to Poincare lemma

Poincare lemma says if a smooth $p$-form $\omega$ is closed, then $\omega$ must be exact. Let's put it in another way, it says the solution of $d\omega=0$ is $\omega=d\eta$ for some $(p-1)$-form ...
3
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1answer
104 views

On the definition of the exponential map

The exponential map on a manifold $M$ is defined at a point $ p\in T_p(M)$ as $$exp_p:T_p(M)\rightarrow M \\ exp_p(v)=\gamma_v(1) $$ where $\gamma_v$ is the constant speed geodesic with initial ...
10
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2answers
312 views

Are closed geodesics the prime numbers of Riemannian manifolds?

I wonder to what extent one can support the analogy that primitive closed geodesics are the prime numbers of Riemannian manifolds? ("Primitive": traced once, as opposed to $m$-fold for $m \ge 2$.) In ...
3
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1answer
73 views

Where does a great circle intersect the hypersurface $X_1^2 + X_2^2 = \delta^2 < 1$ of the unit sphere?

I have reduced a small problem that I am working on (to do with 2-dimensional minimal cones in arbitrary codimension) to some elementary spherical geometry, which I cannot easily manage or visualise. ...
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2answers
386 views

How to prove a curve is a plane curve

There is a regular parametric curve $r(t)$ in $\Bbb{R}^3$,and $r'(t)\bullet a=0$,where $a$ is a fixed vector in $\Bbb{R}^3$, show that $r(t)$ is a plane curve. Thanks very much