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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32 views

When is the frame bundle of a manifold trivial? [closed]

My question is what are the topological restrictions on a manifold $M$ such that its frame bundle is trivial?
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19 views

Computing of proof of Li-Yau estimate

I try to compute the red line in picture below: \begin{align} \Delta(\partial_tL) +R\Delta L +\partial_t R &=\Delta(Q+|\nabla L|^2)+R(\frac{\Delta R}{R}-\frac{|\nabla R|^2}{R^2}) + \Delta R +R^2 ...
2
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1answer
40 views

What is the skew-symmetric part of the covariant derivative of a one-form?

This is a followup question to here. Let $E \to M$ be a vector bundle with connection $D := \nabla$. Extend $D$ to $E^*$ and $\text{Hom}(E, E)$. Let $E = TM$ here, and suppose that the torsion is ...
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0answers
15 views

covariant derivative of a helicoid

Given a helicoid $S$ parametrized by $x(u,v)=(v\cos(u),v\sin(u),u)$, a point $p=(1,0,0)$ on the helicoid, a tangent vector $v=(2,1,1)$ on $T_pS$ and a tangent vector field ...
4
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1answer
69 views

Why is the image of the implicit function in the implicit function theorem not open?

We have a continuously differentiable function $f$ from $\mathbb{R}^{n+m}$ to $\mathbb{R}^n$, and we find a continuously differentiable function $g$ which maps points from $\mathbb{R}^m$ into ...
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0answers
49 views

Explaining an answer involving tensors

I am sorry to ask this but I was reading this question, because I want to solve that too, the thing is that it has been answered a long time ago and I don't understand the answer given there. ...
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0answers
32 views

Integration of forms on manifolds

If you have a $n$-form $\omega$ on $\mathbb{R}^n$, then $\omega = f \mathop{}\!\mathrm{d}x_1 \wedge \dots \wedge \mathop{}\!\mathrm{d}x_n$ locally. Integrating $\omega$ is easy now - let's assume ...
3
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1answer
54 views

Zeroes of exact differential forms on compact manifold

Let $M$ be a $n$ dimensional compact differentiable manifold. I would like to show that any exact differential form of degree $n$ vanishes at at least one point. I think it is a generalization of the ...
1
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1answer
30 views

Parallel surface

For a regular surface $\mathbf{x} = \mathbf{x}(u,v)$ Define $\mathbf{y}(u,v) = \mathbf{x}(u,v) + t \mathbf{N} (u,v)$ where $\mathbf{N}$ is the unit normal of $\mathbf{x}$ How could I show the ...
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1answer
47 views

Ellipse's farthest point to another point

I am trying to find the farthest and closest points of a ellipse without using any brute force type of coding. The processing power is limited so it should be as pinpoint as possible. I have tried a ...
3
votes
2answers
37 views

Question on ideal triangle and hyperbolic distance

I'm asking a question about a construction due to Thurston. Let's consider a hyperbolic triangle (I'm considering the Poincarè disc model of the hyperbolic plane) and from each one of the three ...
2
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1answer
36 views

Finding the envelope of the family $(x-c)^2+y^2=1+c^2$

I have this family of circles: $(x-c)^2+y^2=1+c^2$. I'm to find the envelope of this family. Going by what I know, I write $$F(x,y,c)=(x-c)^2+y^2-1-c^2=x^2-2xc+y^2-1=0.$$ Then, $$\frac{\delta ...
1
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0answers
45 views

Equivariant flat U(1) bundle on a torus

I am trying to understand equivariant flat $U(1)$ bundles on a torus, $(S^1)^N$. By equivariant, I mean equivariance with respect to the natural action of $(U(1))^N$ on $(S^1)^N$. $G$-equivariant ...
2
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1answer
25 views

Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ iff $i(X)\omega=0$ and $L_X\omega=0$

Let $M$ be connected and let $\pi:M\times N \rightarrow N$ be the natural projection. Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ for some $p-$form $\alpha$ on $N$ if and ...
1
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1answer
20 views

Show that exist a unique field $G$ on $TM$ whose paths are of the form $t\to (\gamma(t),\gamma'(t))$, where $\gamma$ is a geodesic on $M$.

Show that exist a unique field $G$ on $TM$ whose paths are of the form $t\to (\gamma(t),\gamma'(t))$, where $\gamma$ is a geodesic on $M$. My approach: Suppose such field actually exist, consider a ...
1
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0answers
32 views

When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
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26 views

Orientability of Complex Grassmannian

I have seen a statement that complex Grassmannian of any dimension is orientable, while real Grassmannian is orientable iff it is even-dimensional. Is there a way to prove it using elementary means ...
1
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1answer
42 views

How to prove this formula involving the push-forward funtion

Let $f:M^{n} \to N^{m}$, and suppose that $(x,U)$ and $(y,V)$ are coordinate systems around $p$ and $f(p)$, respectively, then I want to prove that $$f_{*} (\frac{\partial}{\partial x_i} \big{|}_p ...
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0answers
11 views

Invariant of support function and support point under parallel translation

Picture below is from the 222 and 220 page of this paper,why the support function and support point is invariant under parallel ?
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1answer
20 views

On a proof that the metric volume form is parallel wrt to the Levi-Civita connection

In the context of (semi-)Riemannian geometry, the following fact is well-known: if a (semi-)Riemannian manifold $(M,g)$ is oriented, then the unique volume form $\epsilon = \mathrm{vol}_g$, induced by ...
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0answers
19 views

homotopy formula for currents

I just discovered what we call the homotopy fomula for currents, and i'm trying to prove it, i only have one difficulty: The situation is as follows: I took a homotopy h between two functions f and g ...
1
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1answer
47 views

Every $\mathcal{C}^1$ manifold can be made smooth?

I heard of a theorem saying that each $\mathcal{C}^k$-manifold with $k\geq 1$ can be made into a smooth manifold, i.e. $\mathcal{C}^{\infty}$ (by restriction of the atlas). However, I cannot find ...
2
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1answer
27 views

Lemma characterizing second fundamental form, do not understand step

Consider an excerpt of a lemma and part of its proof from a Riemannian geometry textbook. Lemma. The second fundamental form is independent of the extensions of $X$ and $Y$; bilinear ...
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1answer
51 views

How to prove this formula involving diferentials

Let $f:M^{n} \to N^{m}$, and suppose that $(x,U)$ and $(y,V)$ are coordinate systems around $p$ and $f(p)$, respectively, then I want to conclude that for $g:N \to \mathbb{R}$ $$\frac{\partial(g ...
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1answer
46 views

Diff. Geometry - prove that a straight line is the shortest length in Euclidean 3-space.

Use the following scheme to prove that in E3 a straight line is the shortest distance between two points. Let X : [a,b] → E3 be a curve, and set x(a) = p, x(b) = q. Hi everyone - I wanted to make ...
2
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0answers
31 views

Comparing Two Statements of the Rank Theorem

I don't think this a duplicate, even though a similar question appears here. Let $m\geq n$ and let $F:\mathbb R^n\to \mathbb R^m$ be a $\mathcal C'$ mapping s.t. rank $F'(x)=r\leq n$ for all $x\in ...
1
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1answer
48 views

Obtaining the Rodrigues formula

On $So(3)$ the algebra of a $3 \times 3$ skew symmetric matrices define Lie bracket $[A,B]=AB-BA$ Consider the exponential map $$EXP: So(3) \to So(3)$$. We have the $So(3)$ matrix ...
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0answers
42 views

How interpret this notation.

Let $f : M \to \mathbb{R}$ be $C^{\infty}$. For $v \in M_p=\pi^{-1}(p)=\{\text{fibre of M at p}\}$ I want to conclude that $$f_{*}(v)=df(v)_{f(p)} \in \mathbb{R}_{f(p)}$$ Question I don't ...
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0answers
32 views

Find the curvature of the metric $ds=\frac{|dz|}{(1+|z|^2)}$ on $\mathbb{C}$

The curvature of the metric $g$ is defined as $$k(z)=-\bigg(\frac{2}{\alpha(z)}\bigg)^2 \partial \bar\partial log \alpha(z)$$ where $\alpha$ is positive and real valued. Also ...
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1answer
18 views

Proof hyperbolic surfaces have two asymptotic directions

I have seen it stated that: For every point in a surface of negative Gaussian curvature, there are exactly two asymptotic directions, i.e ones in which the normal curvature is zero. How can this ...
3
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1answer
23 views

$S^1$ acting on $SO(n+1)/SO(n-1)$ by translations

I'm ready right now in a paper, that $S^1$ acts on $SO(n+1)/SO(n-1)$ by right translations. I thought that a Liegroup $G$ acting by right translations, means that we have a right action $\varphi ...
2
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1answer
73 views

Exercise about Lagrangian submanifolds

I am trying to solve the following exercise: Let $(M,\omega)$ be a symplectic manifold and $L$ a compact Lagrangian submanifold such that $H^{1}(L)=0$. Let $\{L_{t}\}_{t\in(-1,1)}$ be a smooth family ...
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0answers
15 views

Proof that $M_ {\boldsymbol f}$ has a neighbourood diffeomorphic to the product $T^n\times D^n$

I'm reading Arnold's proof of Liouville's theorem and got stuck with the following problem in subsection §50, A. Here the manifold $M_{\boldsymbol f}$ is defined as $\boldsymbol F^{-1}(\boldsymbol ...
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0answers
34 views

Solvable group which is not Virtually nilpotent

What is a example of Solvable group which is not Virtually nilpotent(does not have any nilpotent subgroup of finite index)?
6
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1answer
101 views

How much classical geometry must a geometer know?

From my reading Wikipedia, I understand there are several branches of classical geometry (if the ordering is off, or I'm missing a few things, let me know): Absolute Euclidean Non-Euclidean ...
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0answers
39 views

Definition of Regular Point

In "An Introduction to Manifolds (second edition)" by Louring W.Tu, regular point is defined as: Definition 8.22. A point p in N is a critical point of F if the differential $F_{∗,p}: T_pN ...
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54 views

Why such coordinates are still called “isothermal” in the Lorentz case?

We know that in a two dimensional Riemannian manifold there always is, at least locally, a coordinate system in which $g_{ij} = \lambda^2 \delta_{ij}$ - such coordinates are called isothermal. There ...
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0answers
36 views

An intuitive question about the metric $d(\cdot,\cdot)$ on a complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian manifold. Suppose $\gamma_1,\gamma_2:[0,1] \to M$ are two different segments(i.e.minimizing geodesics) from $p$ to $x=\exp_p(v)$. Suppose further that ...
1
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1answer
83 views

Exercise about cotangent bundle

I am trying to solve the following exercise: Let $Q$ be a manifold, $T^{*}Q$ its cotangent bundle and $\pi:T^{*}Q\rightarrow Q$ the bundle map. Let $\omega$ be the standard symplectic form on ...
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0answers
17 views

References on geodesic convexity in infinite dimensional and compact manifolds

Does anybody know references about geodesic convexity in infinite dimensional and compact manifolds? Thanks in advance.
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3answers
44 views

Reference request: integration of *one*-forms along curves on a differentiable manifold.

Could somebody please direct me to a book/lecture notes with an introduction to integration of one-forms along curves in a differentiable/Riemannian manifold -- preferably leaning more towards ...
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1answer
10 views

Are segment domains closed?

Let $M$ be a complete Riemannian manifold. Its segment domain is defined by: $$ \mathbf{seg}(p)= \{v\in T_pM: \exp_p(tv):[0,1] \to M \textit{ is a segment} \ \ \} $$ (Note: "segment" has many ...
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1answer
30 views

Vector bundle vs Total Space

On page 59 in Lee's "An Introduction to Smooth Manifolds" the author writes, "Let $E$ be a smooth vector bundle over a smooth manifold $M$, with projection $\pi:E\to M$." I thought the vector bundle ...
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24 views

Full definition of Rough Laplacian and induced formal adjoint of covariant derivation?

Can everybody give a good reference for full definition of Rough Laplacian of tensor field and induced formal adjoint of covariant derivation on a riemannian manifold? I find some equivalent ...
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1answer
61 views

Why is a differential equation a submanifold of a jet bundle?

I'm reading The geometry of jet bundles by D.J Saunders and struggle with the definition of a differential equation 6.2.23. on page 203. First of all, Saunders introduces a differential operator ...
3
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1answer
62 views

Proof that this is a smooth manifold

Let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^4$ be defined by $F(u,v) = (u+v, uv, u-v, v^3)$ and let $M = F(\mathbb{R}^2)$. Prove that $M$ is a smooth manifold. The proof that my TA posted online was ...
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0answers
20 views

Confusion surrounding the Koszul-Malgrange theorem

On a recent project, I ran into something which seemed related to the Koszul-Malgrange theorem. According to nlab, the original statement of the theorem is Theorem 1. (Koszul-Malgrange theorem) ...
2
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2answers
33 views

Why image of curvature is a Lie subalgebra?

In the red line of picture below, why it is Lie algebra ? $M_{\alpha\beta}$ is the Lie bracket ? But $M_{\alpha\beta}$ is symmetric . Picture below is from the 216 page of this paper. ...
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24 views

Proof of Hamilton's strong maximum principle.

As picture below, Why $\forall v\in \text{null}(M_{\alpha\beta}), \nabla_iv\in\text{null}(M_{\alpha\beta})\Rightarrow \text{null}(M_{\alpha\beta}) \text{ is invariant under parallel translation}$ ? ...
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27 views

Rank of curvature operator under Ricci flow.

I think under Ricci flow ,the rank of curvature operator does not change by +1 or -1, it will directly change to full rank or zero rank . I want to write it as term paper, but I don't know whether ...