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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Curvature of $K$-invariant connection (principal bundles)

Here is a proposition from Kobayashi & Nomizu's Foundations of Differential Geometry. I don't understand how they obtain the final line of the proof. They write: \begin{align} ...
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26 views

Lie group action from the Lie algebra

want to find the corresponding lifting f the standard U(n) lifting on $C^n$ to $L=C^n \times C$ with hermitian metric $e^(-|z|^2)$. I try to follow the method in Donaldson, and I find if B in u(n) ...
2
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1answer
37 views

Typo in “Intro to Contact Topology” by Geiges, Lemma 1.4.10?

In Introduction to Contact Topology by Geiges, there is a result relating Hamiltonian and Reeb flows for hypersurfaces of contact type in a symplectic manifold. Lemma 1.4.10 $\,$If a codimension 1 ...
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Do two embeddings of a Euclidean space into a higher dimensional one only differ by a diffeomorphism?

Let $d\le n$ and $$f,g\colon\mathbb{R}^d\hookrightarrow\mathbb{R}^n$$ be two smooth embeddings. Is there a diffeomorphism $$\phi\colon\mathbb{R}^n\rightarrow \mathbb{R}^n,$$ such that $$f=\phi\circ ...
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32 views

continuous (smooth) maps and group homomorphism

Consider a topological group $G$ (or smooth Lie group) and a topological space $M$ (or smooth manifold) and a group homomorphism $\phi:G\rightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M, ...
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1answer
14 views

Proof a special extension on the isoperimetric theorem

Let both ends of a string of length $L$ be tied to a stick of length $S$. Among all plane regions enclosed by this contraption, it achieves maximum when the string forms a circular arc. It is noted ...
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86 views

Showing that continuous forms are zero on a $\mathscr{C}^1$ simplex $\Psi$ , if all smooth forms are zero on $\Psi$.

Question: Is the guess below correct? EDIT: There haven't been any responses yet; I wonder if the question needs to be improved somehow... Forms and simplexes are as in Rudin Rudin Principles of ...
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0answers
41 views

Preserving arc length for a family of curves

Sorry if I have formatted things wrong, I have read the tour and browsed around, so I tried my best. I have a one parameter family of curves with the relation: $$\frac{\partial}{\partial \lambda} ...
5
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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
32 views

Open balls in the definition of a Euclidean submanifold

Stroock (Essentials of Integration Theory For Analysis, $\S8.3.4$) defines a $k$-dimensional submanifold of $\mathbb{R}^n$ to be a set $M \subseteq \mathbb{R}^n$ such that for all $x \in M$, there ...
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1answer
43 views

Finding the curvature and normal vector for an arclength curve

I am trying to find the curvature and normal vector for $$\alpha(t) = \left(\frac13 (1+t)^{3/2}, \frac13 (1-t)^{3/2}, \frac{t}{\sqrt2}\right)$$ $$\alpha'(t) = \left(\frac12 (1+t)^\frac12 , - \frac 12 ...
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34 views

Curvature and Torsion explained

Let $\alpha: I \to \Bbb R^3$ be a curve parameterized by arc length $s$, with curvature $k(s)\ne 0$, for all $s\in I$ $$\alpha(s) = \left(a\cos \frac sc, a \sin \frac sc, b\frac sc\right),\quad s\in ...
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0answers
24 views

Formal adjoint of curvature (Yang Mills)

Currently reading a paper on finding solutions to the Yang Mills equation $D^*\Omega=0$, where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. ...
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34 views

Differential Geometry concept verification

If I have a regular parameterized curve: $$\alpha(t)$$ The curvature, $k(t)$, is precisely $\|\alpha''(t)\|$, The normal vector, $n(t)$ is found by looking at $\alpha''(t) = n(t)k(t)$ The binormal ...
1
vote
1answer
17 views

Local extension of a function on an immersed submanifold

Consider the following passage in Spivak's Differential Geometry book: I am having trouble understanding where he says $g = \tilde{g} \circ i$ on $V \cap M_1$. Since $V$ is (I think) supposed to be ...
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31 views

Singularity in Ricci flow vs Ricci soliton

In the paper "The formation of singularity in Ricci flow" Hamilton studied systematically the possible singularities of the flow.My question is why it is important to classify Ricci solitons in order ...
2
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0answers
50 views

$V$-bundles and vector bundles

I am looking for more information on $V$-bundles. They are hard to search for as either vector bundles come up or something like GL($V$)-bundles come up. I am looking for some nice expository ...
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1answer
27 views

Proof of the fundamental inequality of the index form

I am looking for a proof of the fundamental inequality of the index form, which I have seen as references or statements in a lot of sources, but without a proof. This is the statement: Let $M$ be a ...
5
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32 views

Solving a 2nd-order elliptic PDE with non-constant coefficients

I wonder how I can solve the following 2nd-order PDE on the positive semiplane $\{x>0\}$: $$(\partial_x^2+\frac{1}{x}\partial_y^2)\phi=\delta(x-x_0)\delta(y).$$ I notice that the l.h.s. is the ...
5
votes
1answer
61 views

Not all tangents to a plane curve are bitangents

I'm struggling on a question from a previous qualifying exam, and I don't see a clean way to do it. Let $X\subset \mathbb{R}^2$ be a connected, 1-dimensional, real analytic submanifold, not contained ...
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40 views

Level sets on $SU(n)$

Given $G \in SU(n)$, what are the level sets of the function $F(V) = |tr(G^{\dagger}V)|^2$? Can they be written only in terms of abstract linear maps, not in terms of the components of $V$ and $G$ in ...
5
votes
1answer
86 views

Proving that given any two points in a connected manifold, there exists a diffeomorphism taking one to the other

Suppose $M$ be a connected manifold and $x, y \in M$ are two points. Then I'm trying to show that there is a diffeomeorphism $f$ of $M$ that takes $x$ to $y$. Since the set of points for which there ...
3
votes
1answer
39 views

Is it meaningful to take “exterior products” of vector fields?

Let $M$ denote a smooth manifold. I've read that a differential $k$-form is a smooth section of the $k$th exterior power of the cotangent bundle of $M$. However I barely understand what this means, ...
2
votes
1answer
53 views

Scalar product on Lie algebra of compact Lie group [duplicate]

I am studying Differential Geometry and I am facing with a lemma in which there is a step that I do not understand. In particular, let $G$ be a connected compact Lie group, is used "$\langle\ \cdot , ...
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1answer
38 views

Alternative Proof of why Every Manifold is Locally Compact

So while I was solving some problems on differential geometry, I stumbled upon a problem which is to show that every manifold is locally compact. Now, there is a proof for it here, but I was thinking ...
3
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2answers
47 views

Parallel transport along a 2-sphere.

I'm currently learning about parallel transport and connections and we were considering the parallel transport of a tangent vector along a sphere as given in the picture below. From my ...
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0answers
30 views

A detail in the proof orientable manifold admits exactly two orientations

Let $M$ be a real manifold and let $\{(U_\alpha,\phi_{\alpha})\}_{\alpha\in I}$ $\{(V_\beta,\psi_\beta)\}_{\beta\in J}$ be two oriented atlases. Let's define, for $p\in U_\alpha\cap V_\beta$, ...
2
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1answer
32 views

“Winding number”, Chern character and relative signatures of the metric

Anyone answer with good explanation is appreciated. In differential geometry, we discuss about topological quantities like characteristic classes. For example, the first Chern character of some ...
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0answers
16 views

Locus of points on a rotating line ; points differently ordered

A line rotates about a fixed point $O$ with ordered points $P,O,M $, while $ M $ is moving along this line $POM$. Find locus of points $ P ,M $ if $ MP^2- OM^2 = T^2 $ constant for all inclinations ...
2
votes
1answer
57 views

Expressing a differential form in terms of a scalar function

We can express every k-form in the form $ \omega(x) = \sum_Id_Idx_I $ where $ I$ is k-tuple and $ d_I$ is just some scalar function of x. That's entirely understandable for me. But while reading ...
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1answer
268 views

Can the interior of a manifold be orientable but not its boundary?

Suppose $M^m$ is a manifold with boundary. If we are given an orientation for $M$, we can then derive an orientation for $\partial M$ by considering the orientation of $TM$ at $\partial M$ and then ...
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34 views

Show that $1$-form has particular coordinate representation.

I want to show that for a nowhere-vanishing $1$-form $\alpha$ on an $n$-dimensional manifold $M$ we have that $\alpha \wedge d\alpha = 0$ if and only if around every point $p$ there exists a ...
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1answer
23 views

Hyperspheres inside of a Hypercube

How many $4$-dimensional hyperspheres with a diameter of $1$ can fit inside a $4$-dimensional hypercube of length of $2$?
2
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1answer
32 views

Without loss of generality in proof about subspaces in symplectic linear algebra

A linear symplectic space is a 2n-dimensional vector space $V$ with a symplectic two form $\omega.$ On this vector space $V$ is a canonical basis $(e_1,...,e_n,f_1,...f_n)$ with $\omega(e_i,f_j) = ...
0
votes
1answer
32 views

Lie bracket in coordinates

In $\mathbb{R}^2-\{0\}$ we consider the vector field defined by, $$V=-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}$$ I am trying to find all other vector fields $X$ that $[V,X]=0$. My ...
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2answers
301 views

Prove (local) converse to the implicit function theorem

The implicit function theorem tells us that: Given a level set $M^k = F^{-1}(F(p_0))$ of a smooth function $$F: \mathbb{R}^n \to \mathbb{R}^{n-k},$$ where $\operatorname{rk}{(Df)(p)} = n-k$ for ...
3
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1answer
51 views

A question on Stokes theorem for Lipschitz functions

Let $M$ be an oriented compact Riemannian manifold. Let $f$ be a Lipschitz function on $M$, denote $M'\subset M$ be the set on which $f$ is differentiable. On one hand, Stokes theorem works for ...
0
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1answer
31 views

Finding the Frenet frame [duplicate]

I am trying to find the Frenet frame of the following curve: $$\zeta(t)=\left(\frac13(1+t)^{3/2},\frac{1}3(1-t)^{3/2},\frac{t}{\sqrt2}\right)$$ How do I do this? Is there a straightforward way from ...
4
votes
1answer
53 views

Pontryagin class of a wedge product of vector bundles.

Let $E\to M$ be a real vector bundle over a differentiable manifold $M$ and let $p_{1}(E)$ denote its first Pontryagin class. I would like to know if there is any formula allowing to write ...
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How to find coordinates of points on a 2D surface embedded in 3D space

kindly assist with this problem. Given an equilateral triangle in 2D plane (see figure 1) with origin (0,0) at point B, the coordinates of points A and C can be calculated as A(acos60,asin60) and ...
2
votes
1answer
35 views

Derivation of tensor components transformation in tangent space

Might anyone offer a derrivation? My attempt bellow ($ x_{i'}$ is counting through the transformed coordinates) $\displaystyle\frac{\partial }{\partial x_{i'}}= \displaystyle\frac{\partial ...
3
votes
1answer
40 views

Perpendicular Gradients

Suppose $f:\mathbb{R}^2\to \mathbb{R}$ is smooth. Further suppose $\nabla f$ vanishes no where. When is it possible to find a smooth non-singular $g:\mathbb{R}^2\to \mathbb{R}$ satisfying $\nabla ...
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1answer
27 views

Circle determined by three points on a curve tends to the osculating circle

I am stuck on problem 3.3.2 of Differential Geometry of Curves and Surfaces by Banchoff and Lovett. The problem is: Let $\vec{x}(s) \colon I \to \mathbb{R}^2$ be the parametrization by arc length of ...
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1answer
58 views

Curvature of plane parametric curves

What is the neatest way to derive the following formula for the curvature of a parametric curve? $$\kappa =\frac{||y'x''-y''x'||}{(x'^2+y'^2)^{\frac{3}{2}}} $$
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The derivative of signed unit normal is proportional to the curvature

I'm trying to study for one of my exams and the past papers have no solutions. I had to define the signed unit normal and the signed curvature. The signed curvature, $\kappa_s$ being such that ...
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1answer
42 views

Integral curves of vector fields on closed surfaces

If we have a vector field on a boundary less and compact 2-manifold, which is neither a gradient nor a harmonic, does that imply its integral curves are closed?
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0answers
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A nonplanar closed curve such that the plane curve with the same curvature as function of the arclength is not closed

Find a non plane, closed curve such that the plane curve with the same curvature as function of the arclength is not closed. Been thinking a lot in this problem and haven't got a clue. Any ...
3
votes
1answer
47 views

Möbius band as line bundle over $S^1$, starting from the cocycles

The professor asked us to construct a non-trivial line bundle over $S^1$ by giving an open cover of $S^1$ and the cocycles. My idea was to take as open cover $U_1:=S^1\setminus\{0\}$ and ...
2
votes
2answers
83 views

What is topologically the set of all straight lines in $\mathbb{R}^d$? More structures on it?

If we consider the set of all straight lines in $\mathbb{R}^d$, then what is it topologically? If it's topologically 'nice' i.e. a manifold, probably we could put a smooth manifold structure on it ...
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1answer
36 views

Rank function can change upwards but not downwards: need intuition

Let $f: \mathbb R^n \to \mathbb R^m$ be some smooth map and $J_f$ its Jacobian. Say $x \in \mathbb R^n$ is such that $$ \operatorname{rank}{(J_f (x))} = p$$ Then there exists a neighbourhood of ...