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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diferential equation system differential operators method

$x'-3x+2y=t$ $y'+2x=e^t$ it is asked to solve by the mentioned method $\Delta(D)=D(D-5)$ $\Delta_1=1-e^t$ $\Delta_2=-2t-2e^t$ $yD^3(D-5)(D-1)=0$ $xD^2(D-5)(D-1)=0$ When solving for the ...
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39 views

What is the definition of a properly-immersed arc?

I am going over a paper in which a theorem includes a condition that certain arcs are "properly-immersed geodesic arcs" in a surface $\Sigma$ with nonempty boundary. I have done a search, but I have ...
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36 views

The change of parameter of a regular curve is a diffeomorphism, and preserves the length

Let $C$ be a regular curve and let $\alpha:I\subset\mathbb{R}\to C$, $\beta:J\subset\mathbb{R}\to C$ be two parametrizations of $C$ in a neighborhood of $p\in\alpha(I)\cap\beta(I)=W$. Let ...
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62 views

Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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54 views

Is the tangent-cotangent isomorphism orientation preserving?

Consider $(M,g)$ a Riemannian manifold. Let's define $\varphi : TM\rightarrow T^{\ast}M$ by $\varphi(p,v):=(p,g(v,.))$, for $p\in M$ and $v\in T_{p}M$. Here, $TM$ stands for the tangent bundle and ...
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33 views

What's wrong in this prop about volume form if we drop “oriented”?

I was studying Prop 15.29 from Lee's Introduction to Smooth Manifold and I asked myself what's wrong with this proof if we drop the oriented assumption. I know that I'd came up with a non zero ...
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85 views

Proving a nowhere vanishing vector field on 2D manifold implies $TU\cong M\times S^1$

So, I am trying to solve the following problem. Suppose you have a nowhere zero smooth vector field on a 2 dimensional oriented compact manifold. Prove that the unit tangent bundle $TU$ is ...
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42 views

Topological covering + local diffeomorphism gives smooth covering

I got stuck at some point while working on this part of an exercise from Lee's Introduction to Smooth Manifolds, 2nd edition. The part which I am stuck on is to prove (one of the directions of ...
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21 views

global function defining a $\mathcal{C}^1$ submanifold

Let $M$ be a $\mathcal{C}^1$ submanifold of dimension $1$ of $\mathbb{R}^2$. Then for each $x\in M$, there is a neighbourhood $U$ of $x$ and a $\mathcal{C}^1$ function $f:U\to \mathbb{R}$ such that ...
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78 views

Covariant derivative and geodesic

Let $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be a surface patch. Then if we have two vector fields $$X = \sum_i \xi^i \frac{\partial f}{\partial u^i}$$ and $$Y = \sum_i \eta^i ...
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122 views

Why is first fundamental form considered intrinsic

I am reading Kuhnel's differential geometry book, and in chapter 4, it says that "intrinsic geometry of a surface" can be considered to be things that can be determined solely from the first ...
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1answer
75 views

Prove that the curvature of $\gamma$ is $\frac{\kappa_{\alpha}}{\sin^2\theta}$

Let $\alpha:I\to {\mathbb R}^3$ be a cylindrical helix with a unit vector $u$ such that $u\cdot T_{\alpha}$ is a constant for all $t\in I$. For $t_0\in I$, the curve ...
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3answers
157 views

Prove: $\kappa^2v^4=|\alpha^{''}|^2-(\frac{dv}{dt})^2.$

Given a regular curve $\alpha:\mathbb R\to {\mathbb R}^3$, Prove: $$\kappa^2v^4=|\alpha^{''}|^2-\left(\frac{dv}{dt}\right)^2.$$ ,where $\kappa$ is the curvature, $v$ is the rate of change of ...
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49 views

Compute the contraction of a 1-form with a vector field

Question: Let $\alpha$ be the $1$-form on $\mathbb{R}^3$ given by $\alpha=zdy-ydz$ and let $\mathbb{X}$ be the vector field on $\mathbb{R}^3$ given by $\mathbb{X}=(0,y,-z)$. Compute ...
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56 views

Hodge star operator and volume form, basic properties

let $(M,g)$ be an oriented Riemannian manifold. Let $*$ be the hodge operator, I want to prove that $$*\mathrm{vol}_g =1$$ where $\mathrm{vol}_g$ is the associate volume form $\sqrt{g} e^1\wedge ...
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1answer
46 views

Is $M \times (0,\infty)$ a manifold of bounded geometry?

If $M$ is a compact Riemannian manifold, is $M \times (0,\infty)$ a manifold of bounded geometry? I think it is, since $M$ is compact and $(0,\infty)$ is simply flat.
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2answers
79 views

Morse function on $\mathrm{SO}(n)$

I would like to prove that the following function is a Morse function, $F : \mathrm{SO}(n)\to\mathbb{R}$ $$A=(a_{ij})\mapsto\sum_{i=1}^na_{ii}\lambda_i$$ with $0<\lambda_1<...<\lambda_n$. ...
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1answer
54 views

Euler characteristic of the closed unit ball

I would like to calculate the Euler-Poincaré characteristic of the closed unit ball $B$ by the de Rham cohomology, the Poincaré-Hopf theorem and Morse theory. de Rham cohomology: since $B$ is ...
3
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1answer
57 views

How to determine whether a differential $1$-form is globally welldefined?

This is a question that occurred after working on finding a generator of the first de Rham cohomology group of the torus. It was pointed out to me that the differential $1$-form $$ dx + dy$$ was ...
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1answer
29 views

Heat kernel property

We define the heat propagator on a Riemannian manifold $M$: $$e^{-t \Delta_g}: L^2(M) \rightarrow L^2(M)$$ $$e^{-t \Delta_g} f(x) = \int_M p(x,y,t) f(y) \,dV(y)$$ where $p(x,y,t)$ is the ...
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2answers
90 views

Applying the Frobenius theorem to a decomposable 2-form

So I have the following problem: Suppose $\omega=\phi \wedge \theta$ is a closed decomposable 2-form on $M$ a manifold (decomposable just means it can be written as a wedge of 1-forms). Suppose $p\in ...
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3answers
69 views

Generators of $H^1(T)$

Let $T$ denote the torus. I am working towards an understanding of de Rham cohomology. I previously worked on finding generators for $H^1(\mathbb R^2 - \{(0,0)\})$ but then realised that for better ...
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43 views

Proof Transverse Submanifolds

Let $f:M\rightarrow\mathbb{R}^p$ differentiable and $N$ submanifold of $\mathbb{R}^p$. Show that for all $\epsilon>0$ exist $v\in\mathbb{R}^p$ whit $||v||<\epsilon$ such that $f(x)=f(x)+v$ is ...
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26 views

Right Veering Property of elements in MCG(S)

Let h be an element of MCG(S), the mapping class group of a surface S. I was going over : I was going over :Geometric Intuition for "Right-Veering" Property of $f$ in MCG(S)? Where a p.e ...
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1answer
23 views

Derivation at a point factors through derivation on algebra of germs

Let $M$ be a manifold and $C_x^{\infty}(M,\mathbb R)$ be the algebra of germs of smooth functions on $M$ at $x$. A derivation of $C^{\infty}(M,\mathbb R)$ at a point $x$ is a linear map ...
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67 views

Completeness implies geodesic completeness, a more conceptual way?

We know from Riemannian geometry that for Riemannian manifolds, completeness and geodesic completeness are equivalent, which is usually a consequence of Hopf-Rinow theorem. However, I'm considering a ...
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80 views

Beltrami-Enneper theorem

Given that geodesic torsion $ \tau_g $ and Gauss negative curvature $K$ are constant along a line on a surface show that the line must be asymptotic, i.e., must have a vanishing normal curvature ...
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52 views

Show $k$-form/chain identity

Let $\omega$ be a closed $k$-form on $\mathbb{R}^n$ and $c:I^k\rightarrow\mathbb{R}^n$ a $k$-cube on $\mathbb{R}^n$. Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$ with flow $\Phi_t$. Show that ...
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58 views

Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...
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1answer
58 views

Using Poincaré lemma to find the generator $H^1$ of $\mathbb R^2 -\{(0,0)\}$?

I am working towards an undertanding of de Rham cohomology. For this reason I am trying to find generator(s) of $H^n_{dR}(\mathbb R^2 -\{(0,0)\})$ and currently I am working on the case $n=1$. I ...
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2answers
64 views

Remember the Christoffel symbols

This might be a little bit different from what is asked normally on this page, but does anybody here know a good way to remember the definition of the Christoffel symbol? \[ \Gamma^k_{ij} = \frac 12 ...
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25 views

Uniqueness of backward heat equation on closed manifold with given initial data

Suppose we have a closed (compact without boundry) Riemannian manifold $(M,g)$, do we have uniqueness (assuming solutions exist on $[0,T]$ ) for the backward heat equation $$\frac{\partial f ...
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34 views

Showing $\dim T_0 ℝ^n = n$ using a derivation definition for the tangent space.

I’m trying to (re-)prove that $\dim_ℝ \mathrm{Der}_ℝ(C^1(ℝ^n)) = n$, where $$\mathrm{Der}_ℝ(C^1(ℝ^n)) = \{δ\colon C^1(ℝ^n) → ℝ;~\text{$δ$ is a $ℝ$-linear derivation}\},$$ and the ...
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1answer
47 views

How is defined the notion of $C^1$-close submanifolds?

The question is already in the title. Reading some papers, I have find statements like the following one, with no reminder about the notion of $C^1$-closedness, and without references for further ...
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1answer
56 views

Every possible choice of Christoffel symbols generate a valid connection

Does every possible choice of Christoffel symbols generate a valid connection? Or is there some restriction on them?
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2answers
68 views

Actual Classification re Nielsen-Thurston Theorem (how to)?

according to Nielsen -Thurston Classification: http://en.wikipedia.org/wiki/Nielsen%E2%80%93Thurston_classification If $S$ is compact and orientable surface, then any homeomorphism is isotopic to ...
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1answer
87 views

Understanding de Rham cohomology: geometrically speaking, when is a smooth function closed

On Wikipedia the de Rham cohomology groups are defined to be the cohomology groups of the de Rham cochain complex (equivalence classes of differential $k$-forms). By this definition the zeroth de ...
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1answer
56 views

Length of a Curve and Area of Torus

Consider a parametrised surface (the torus of revolution), $$ M(u, v) = ((a + r \cos u) \cos v, (a + r \cos u) \sin v, r \sin u) $$ where $a > r > 0$. a) Calculate the arc length of the ...
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Reparametrizations of curves by nondecreasing changes of parameters have the same length

On p. 45 of the book by Burago/Burago/Ivanov on Metric Geometry, it is claimed that "it is easy to see that all parameterizations of a curve have equal lengths". How do I prove this? If $\gamma\colon ...
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33 views

normal exponential map on Riemannian manifolds

where can I find information about normal exponential map on Riemannian manifolds? please introduce some books.
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1answer
79 views

Inductively prove that $L_{\mathbb{X}}i_\mathbb{Y}=i_{[\mathbb{X},\mathbb{Y}]}+i_\mathbb{Y}L_{\mathbb{X}} $

Let $\mathbb{X}$, $\mathbb{Y}$ be vector fields on $U \subset \mathbb{R}^n$. Prove that $$L_{\mathbb{X}}i_\mathbb{Y}=i_{[\mathbb{X},\mathbb{Y}]}+i_\mathbb{Y}L_{\mathbb{X}} $$ using induction. Assume ...
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24 views

uniform constant and radius bound for poincare type inequalities On compact manifolds

I have a clarification question. If we have a Riemannian compact manifold $M$, then there exists constants c and $r_{0}$ such that for any radius r < $r_{0}$ we have $$ \bigg(\frac{1}{|B_{r}(x)|} ...
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53 views

Show that $\Psi_{t*}\mathbb{X}-\Phi_{t*}\mathbb{X}=\mathbb{X}-\mathbb{Y} $

Let $\mathbb{X},\mathbb{Y}$ be vector fields and let $\Phi_t$ denote the flow of $\mathbb{X}$. Given that $\displaystyle \frac{\partial}{\partial t}\Phi_{t*}\mathbb{Y} ...
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24 views

Find a complex-valued $g(u,v)$ such that $L_\mathbb{Y}g=img$

Let $F$ be a diffeomorphism between open $U$ and $V$ in $\mathbb{R}^n$. Let $\mathbb{Y}$ be a vector field on $V$ and $\omega$ a $k$-form on $V$. Given the identity ...
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40 views

Show that $\int_b d\omega=0$ where $b(s,t)=\Phi_s(c(t)) $

Let $c:[0,1]^k \rightarrow \mathbb{R}^n$; $t \mapsto c(t)$ be k-cell with $k < n$. Let $\mathbb{Y}$ denote a vector field on $\mathbb{R}^n$ with flow $\Psi_s$. Define a $(k+1)$-cell $b:[0,1]^{k+1} ...
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1answer
44 views

I need help understanding this derivation of the geodesic equation.

I'm having a hard time seeing what 'standard techniques' from calculus yield equality $(7.10)$? The proof is : This is from Schaum Series' Tensor Calculus by David C. Kay. We're using this text for ...
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1answer
46 views

mutually transverse embedded submanifolds, natural bundle surjections, direct sum, isomorphism

Let $N$ be a manifold and let $M_1, \dots, M_n \hookrightarrow N$ be mutually transverse embedded submanifolds, so $M = \cap M_i$ is an embedded submanifold of $N$ with $\text{T}_m(M) = \cap T_m(M_i)$ ...
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1answer
42 views

Breaking down what the definition of an affine connection says

In his book Riemannian Geometry, Manfredo do Carmo defines an affine connection as follows: Let $\mathcal{X}(M)$ denote the set of all vector fields of class $C^{\infty}$ on $M$. Let ...
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2answers
24 views

Show that the surface $x^2+y^2=x$ using $\theta \space and \space z$ can be parametrised by $(\cos^2(\theta), \cos(\theta) \sin(\theta), z)$

I really have no idea how to do this: $x^2-x+y^2=0$ looks like it can be a circle given by: $(x-\frac{1}{2})^2+y^2=\frac{3}{4}$ mostly $x=r\cos(\theta) \space and \space y=r\sin(\theta)$ work as ...
2
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1answer
108 views

A non-vanishing one form on a manifold of arbitrary dimension

So the problem I have is: Let $\theta$ be a closed 1-form on a compact Manifold M without boundary. Further suppose that $\theta \neq 0$ at each point of M. Prove that $H^{1}_{dR}(M)\neq 0$. The ...