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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408 views

Positive definiteness of Fubini-Study metric

Define the Fubini-Study metric $$g_{i\overline{j}} = \frac{\delta_{i\overline{j}}(1+|\boldsymbol{z}|^2)-\overline{z}^jz^i}{(1+|\boldsymbol{z}|^2)^2} $$ for $i,j=1,\ldots,n$ and $z_i$ complex variables ...
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49 views

A curve is contained in a hyperplane if and only if $\kappa_{n-1} = 0$

I'm trying to prove that a curve $\gamma$ in $\mathbb{R}^n$ is contained in a hyperplane if and only if $\kappa_{n-1} = 0$, where $$\kappa_{i} = \frac{\langle e_{i}',e_{i+1}\rangle}{\| \gamma' \|}$$ ...
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54 views

Divergence Operator

How do you simplify such operators: $$(\vec{a}\cdot\nabla)\vec{b}$$ I would appreciate any reference/name regarding this so I can try to understand this.
0
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1answer
32 views

Is a cylinder a Lipschitz domain?

I'm wondering if the domain $(0,T)\times \Omega$ is a Lipschitz domain ($T$ is a positive real number), provided that $\Omega$ is an open bounded subset of $\mathbb{R}^n$ with Lipschitz boundary, and ...
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2answers
175 views

Is there any standard N-sphere that has non-trivial first Pontryagin class?

I am wondering if there is a standard $N$-sphere that has non-trivial first Pontryagin class on its tangent bundle $TS^n$and frame bundle $FS^n$? I know that only $S^4$ has non-trivial $H^4(S^n, R)$ ...
4
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1answer
117 views

The degree of every smooth map $\mathbb{R}^n \to \mathbb{R}^n$ is one…

Let $\varphi : M^n \to N^n$ be a proper smooth map between two connected smooth manifolds. Then $\varphi$ induces a linear map $\varphi^* : H_c^n(N) \simeq \mathbb{R} \to H_c^n(M) \simeq \mathbb{R}$ ...
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71 views

Connection on Submanifold using given connection on ambient manifold

This result features in Hicks' Notes on Differential Geometry book.The theorem states that given $C^\infty$ fields X and Y on a submanifold M, we have $$\bar D_X Y=D_X Y+V(X,Y)$$ where $\bar D$ is a ...
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1answer
67 views

Support of the pullback of a function

Let $F: N → M$ be a $C^∞$ map of manifolds and $h: M → \mathbb R$ a $C^∞$ real-valued function. Prove that $supp F^*h \subset F^{-1}(supph)$. I study the problem and I believe that first i need prove ...
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1answer
26 views

Verifying smoothness of a specific 1-form

I'm having trouble understanding the following: My initial thought was that the 1-form is "obviously" smooth, since the coefficient functions are smooth. But then why would the author say that the ...
0
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1answer
66 views

Reparameterize a path $C^1$ to a path $C^{\infty}$

Let $\large \alpha:I\longrightarrow\mathbb{R}^n$ be a path of class $\large C^1$ We can reparameterize $\large \alpha$ such that $\large \beta= \alpha\circ h \in\large C^\infty$ ? Where $\large ...
1
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1answer
35 views

Smooth Manifolds with finite dimension

Let $M_1,M_2$ and $N$ be manifolds of dimension $m_1,m_2$ and $n$ respectively. Prove that the map $(f1, f2): N → M_1 × M_2$ is $C^∞$ off $f_1$ and $f_2$ are $C^∞$. Can you help me please.
9
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2answers
474 views

Applications of information geometry to the natural sciences

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
0
votes
1answer
85 views

Unit speed curves and Frenet frames

Let $\alpha(s)$ and $\beta(s)$ be two unit speed curves and assume that $\kappa_{\alpha}(s)=\kappa_{\beta}(s)$ and $\tau_{\alpha}(s)=\tau_{\beta}(s)$, where $\kappa$ and $\tau$ are ...
0
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1answer
45 views

problem of differential equations

Considering a family of curves $k(x,y,\lambda)=0$ defined in a domain $\omega$ of $R^2$ with $\lambda$ real, I have to calculate the differential equation of the curves intersect those under a ...
2
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1answer
78 views

Intersections of connected sets with piecewise smooth boundaries

Suppose you have two connected sets $S_1$ and $S_2$ in $\mathbb{R}^n$ with piecewise smooth boundaries, and whose intersection $S=S_1 \cap S_2$ has positive Lebesgue measure. Will the sets $S$, $S_1 ...
1
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1answer
294 views

Diffeomorphism and determinant of Jacobian

I don't remember where I read it and if I remember it correctly but does the following hold true? If $M,N$ are two (smooth?) surfaces and $f: M \to N$ is a homeomorphism such that $det(J_f)$ (the ...
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0answers
59 views

question about regular mapping in elementary differential geometry by Oneill

I am looking at Oneill elementary differential geometry section 4.2 Patch Computations. In example 2.4, parametrization of a surface of revolution, it says Suppose that $M$ is obtained by revolving ...
2
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1answer
75 views

Radial geodesics in a graph of a function

I'm trying to figure out how to prove the following claim: Suppose that $S$ is the graph of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ and every plane containing the $z$-axis intersects $S$ ...
17
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1answer
749 views

When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
0
votes
3answers
241 views

A little confusion about compactness and connectedness

This question may be a bit simple or even naive for some people but it indeed confuses me for a long time. Thank you all if you provide any explanation. I know concepts: compactness means any open ...
4
votes
1answer
151 views

Defining a quotient manifold with gluing

I'm trying to find conditions on the gluing map between two manifolds so that the quotient space will be a smooth manifold, and the inclusion map will be a diffeomorphism. Specifically, Suppose ...
3
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1answer
79 views

differential calculus in $\mathbb{R}^\mathbb{N}$?

is it possible to define the derivative of a function of countable variables? I found differential calculus of function with a finite number of variables, or differential calculus in Banach spaces ...
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2answers
85 views

What is the geometric meaning of the number of independent derivatives of $\gamma$?

Let $\gamma:I \to \mathbb{R}^n$ be a curve. I want to see, what is a geometric meaning of the number of independent derivatives of $\gamma$. I guessed it is it's dimension but it was not. Can you help ...
3
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1answer
1k views

Differential Geometry Video Lectures

I know there's a similar question here, however since what I found there wasn't what I was looking for I thought on creating a new question. I'm studying Differential Geometry through Spivak's book "A ...
3
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1answer
70 views

Locally euclidian but not topological manifold

I'm having trouble solving one part of one of the initial exercises of the classic Boothby book "An Introduction to Differentiable Manifolds and Riemannian Geometry" (exercise I.3.1). To be more ...
0
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2answers
211 views

unit speed curves and frenet serret

Let us assume that $\alpha(s)$ is a unit speed curve with $\kappa > 0$. I'm trying to find the vector function $w(s)$ such that $$T' = w \times T,\quad N' = w \times N,\quad B' = w \times B.$$ I ...
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1answer
267 views

The derivative of the inclusion map is the inclusion map of tangent spaces.

Let $X$ and $Y$ be smooth manifolds, let $i:X\to Y$ be the inclusion map, prove $di_x$ is the inclusion map from $T_x(X)$ to $T_x(Y)$. I know this is pretty basic, but can someone show me how to do ...
0
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1answer
87 views

Writing inner product of Hessians as a sum of inner products.

Given an orthonromal frame $\{e_1,...,e_n\}$ on a Riemannian manifold $M$ and two functions $f$ and $g$ on $M$. Can we write the inner product of the Hessians of and and $g$ in the following way $$ ...
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2answers
812 views

Commutation of covariant derivative of functions

Le$f$ be a smooth function on a Riemannian manifold $M$. My questions are: a) If $\nabla_i f$ is a function, why is not true that $\nabla_j\nabla_k\nabla_if=\nabla_k\nabla_j\nabla_if$? This question ...
1
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2answers
61 views

Please checking to find an arc-length reparametrization

Find an arc-length reparametrization of $$c(t)=\langle \cos t+t\sin t, \sin t-t\cos t\rangle$$ for $t\in [\pi, 3\pi/2]$ solution trial: $$c'(t)=\langle -\sin t+\sin t+t\cos t, \cos t-\cos t+t\sin ...
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0answers
25 views

Determine the direction of given parametrization.

I saw an example, which I posted below. First of all, I understand how to show paramtrized curve but I dont understand how to determine the direction of the parametrization. For example, how can ...
2
votes
0answers
101 views

Geodesic on a Reimannian manifold with a random metric tensor

Given a metric tensor $g_{\mu\nu}$ on a Riemannian manifold, it's possible to write the geodesic equations using: $$\frac{d^2x^a}{ds^2} + \Gamma^{a}_{bc}\frac{dx^b}{ds}\frac{dx^c}{ds} = 0$$ where: ...
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0answers
97 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. ...
1
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1answer
247 views

Geometric problem-differential equations

I need to solve the following problem: Consider the stright lines that pass through origin. Find the equation of the trayectories that intersect those straight lines at a constant angle w (use polar ...
1
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1answer
240 views

Arc length parameterization lying on a sphere

Show that if $\alpha$ is an arc length parameterization of a curve $C$ which lies on a sphere of radius $R$ about the origin then $$R^2 = ...
1
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1answer
110 views

Instrinsic definition of concave and convex polyhedron

Is it possible to distinguish a concave polyhedron from a convex one by mesurements made only on its surface, without a reference to the 3d space around it?
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0answers
29 views

Higher Order Torsion

Define an k-Torsion as a measure of how much a parametrically defined curve $x(t)$ where $t$ is a real scalar and $x$ is a vector in $R^n$ deviates from the locally encapsulating k-dimensional ...
0
votes
1answer
52 views

Why is a vector space equal to its tangent space for any point?

I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$. If $x\in V$, then since $V$ ...
7
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0answers
137 views

Translating a passage of a paper by L. Bérard Bergery

I am currently studying the following paper on Einstein manifolds: L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d'Einstein, Inst. Elie Cartan, Univ. Nancy №6, 1-60 (1983). I have ...
7
votes
2answers
1k views

Cartan's magic formula

A possible proof of Cartan's magic formula $$L_X = i_X \circ d+d \circ i_X$$ is to follow the steps: Show that two derivations on $\Omega^{\bullet}(M)$ commuting with $d$ are equal iff they agree on ...
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0answers
20 views

Functions from $[E_8]$ to $E_8$

Let $f: [E_8] \to [E_8]$ be a function between 4-manifolds with intersection form $E_8$. What we know (due to Rocklin) is that $[E_8]$ can't have any smooth structure. Questions: Is it true for all ...
5
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2answers
342 views

Coordinate independence of geometrical objects.

I am still trying to get a good grasp on the motivations behind various concepts in Differential Geometry. But I am struggling to come to terms with how certain concepts have this added attribute of ...
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0answers
127 views

Surface orientation

Let $S_{1}$ and $S_{2}$ be two oriented surfaces ($N_{1}$ and $N_{2}$ their normal fields, respectively). We say that a local diffeomorphism $f$ : $S_{1}$$\rightarrow$$S_{2}$ preserves orientation if ...
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56 views

Why is $f:\mathbb{R}\to S^1$ $f(t)=(\cos(t),\sin(t))$ a local diffeomorphism?

An example in my book says that $f:\mathbb{R}\to S^1$ defined by $f(t)=(\cos(t),\sin(t))$ is a local but not global diffeomorphism. By the inverse function theorem, $f$ is a local diffeomorphism if ...
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votes
1answer
420 views

$dxdydz \to -r^2\sin(\theta)\sin(\phi+\theta)dr d\phi d\theta$?

So I got this answer $-r^2\sin \theta\sin(\phi+\theta)dr d(\phi)d(\theta)$ which I think is wrong because I googled it and it must be $-r^2\sin\theta dr d\phi d\theta,$ but $\sin(\phi+\theta$) clearly ...
3
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2answers
118 views

Equivalent definitions of manifolds

From Lee's Introduction to Smooth Manifolds, p.3: Question Concerning the exercise; what if there is a point $x$ in our manifold $M$ such that it has a neighborhood $N$ that is homeomorphic to ...
2
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1answer
124 views

Lift of a diffeomorphism of the Torus

I'm trying to prove the following formula. Suppose to have $p:\mathbb{R}^{d}\rightarrow\mathbb{T}^{d}$ the canonical projection of the real d- dimensional space in to the d-dimensional torus, and ...
3
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0answers
89 views

Module of smooth vector fields

I want to show that the module of smooth vector fields is a free module over the ring of infinitely differentiable functions on some open subset of Euclidean space. I understand how to prove this from ...
0
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1answer
107 views

Geodesic speed not working

From what I know the geodesic speed is equal to unity. I've made this program to plot geodesics along a surface. The surface is embedded in 3d Euclidean flat space: $$ x=r*cos(\phi) $$ $$ ...
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3answers
868 views

Pushforward of a vector field

Can someone help me with that ? We define $\phi:=(\phi^1,\phi^2):\Omega\subset\mathbb{R}^2\to\phi(\Omega)$ with $\Omega$ such that $\phi$ is a diffeomorphism by ...