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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334 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
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59 views

Confused about intuition behind Lie derivative

I'm trying to fix my intuition behind $\mathcal L_X T$, where $T$ is any tensor field. I'd prefer explanations that are not along the lines of $\mathcal L_XY=[X,Y]$ (I'm not sure how this extends to ...
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1answer
27 views

Basic question about Beltrami differentials

my question must be terribly naive, but I'm stuck right now and don't know how to proceed.. Let $X,Y$ be Riemann surfaces and $f:X\rightarrow Y$ an orientation preserving diffeomorphism. It is known ...
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1answer
24 views

Degree of map on $U(n)$ and roots in $U(n)$

Recently I went to a talk of A.Thom in which he sketched a proof of the fact that the groups U(n) satisfy the Kervaire-Laudenbach conjecture. At some point in the proof you have to argue that the map ...
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$V$ is $C^1$ and $V(x_0)=0$ and $ \nabla V $ is not zero $\{ x : V(x)= c \}$ is a surface with no edge around $x_0$

I am studying lyapanov second method in stablity theory of ODE. I have encountered a geometric lemma which says the following: Assume $ V:\mathbb R^n \to \mathbb R$ is a $C^1$ and $x_0 \in \mathbb ...
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1answer
31 views

Question on meaning of notation

What does the following mean in context to Differential Geometry? The book I am reading uses it without explanation. $${\left. {Df\,} \right|_u}(V)$$
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intersection of normal lines converge to a point

I have some difficulty working this out. Let $\alpha: I\rightarrow R^2$ be a regular parametrized plane curve(arbitrarily parameter), and define n=n(t) and k=k(t). Assume k(t) is not 0 for $t\in I$. ...
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20 views

Definition of $\mathcal{R}$-equivalence

I have a question about the definition of $\mathcal{R}$-equivalence. In this book the definition is given as follows: Let $U_i$ be open subsets of $\mathbb R$ and $t_i \in U_i$ ($i=1,2$). We say ...
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37 views

Volume form on $(n-1)$-sphere $S^{n-1}$

Let $\omega$ the (n-1) form on $\mathbb{R}^n$ $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n}$$ Show that the restriction of $\omega$ to $S^{n-1}$ in ...
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65 views

Which Lie groups are also symmetric spaces?

I've scanned some of the literature on this, but couldn't find an answer to the following simple questions (probably because I'm not an expert): Q1: Let G be a Lie group with a left-invariant metric. ...
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1answer
36 views

Non zero derivative

Let $p,q,x,y:(a,b)\to\mathbb{R}$ be $C^1 ((a,b))$ functions. Knowing that the function: $F_v:(a,b)\times (a,b)\to\mathbb{R};\ v\in (0,1),\ F_v ...
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1answer
30 views

Poincaré invariant corresponds to area?

In this paper(equation 34) click me it says that if we have a symplectic egg, then the area of a $(x_i,p_i)$ clone intersected with the egg is given by the poincaré invariant $\int p dx = \int_0^{2 ...
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1answer
29 views

proof of Monge theorem in differential geometry

I want to proof theorem below (Monge) Show a curve on a surface is a line of curvature iff the surface normal along the curve form a developanle surface . Can some one help me. A hint or ...
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1answer
49 views

$\omega$ is $1$-form on $S^1$.

Let $h: \mathbb{R} \to S^1$ be $h(t) = (\cos t, \sin t)$. How do I show that if $\omega$ is any $1$-form on $S^1$, then$$\int_{S^1} \omega = \int_0^{2\pi} h^*\omega?$$
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1answer
16 views

Local representation of a submanifold as a graph over the tangent plane

I'd like to verify the following statement, which intuitively seems quite reasonable, by a rigorous proof: Let $M \subset \mathbb{R}^D$ be a $d$-dimensional $C^1$ submanifold embedded in ...
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32 views

Proving Sard's theorem

Theorem: (Sard) Let $f:U\to \mathbb R^p$ be a smooth map with $U$ open in $\mathbb R^n$, and let $C$ be the set of all critical points of $f$. Then $m([f(C)]=0$ where $m$ is the Lebesgue measure of ...
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1answer
48 views

Function never equal to 0

Let $p,q,x,y:(a,b)\to\mathbb{R}$ be $C^1 ((a,b))$ functions. Knowing that for $(u,v), (\tilde{u},\tilde{v})\in (a,b)\times (0,1)$ we have that: $$\begin{cases} ...
2
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1answer
36 views

Is level set near a maximum value a circle?

Let $f$ be a $C^2$ function defined on $[0,1] \times [0,1]$. Let $0 \leq f(x) < 1$ on $[0,1] \times [0,1] \setminus \left(\frac{1}{2},\frac{1}{2}\right)$ and $f(\frac{1}{2},\frac{1}{2})=1$. It has ...
2
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1answer
139 views

What does it mean to “calculate in local coordinates” on a manifold?

In differential geometry textbook one sometimes reads "calculating in local coordinates, we obtain..." What does this expression mean? Say, $M$ is a smooth manifold and $h$ is a function on $M$; what ...
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1answer
25 views

Holonomy reduction from constant spinors

Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin, manifold, and let us denote by $S$ the corresponding spinor bundle. The Levi-Civita connection $\nabla$ on $(M,g)$ lifts to a unique spin ...
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37 views

Exercise 3.3 Riemannian Manifolds an Introduction to Curvature

STATEMENT: Let $\gamma(t)=(a(t),b(t)),t\in I$(an open interval), be a smooth injective curve in the $xz$-plane, and suppose $a(t)>0$ and $\dot{\gamma}(t)\neq 0$ for all $t\in I$. Let $M\subseteq ...
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1answer
27 views

Enneper surface is not injective

I'm having trouble proving the following statement: $x(u, v) = (u − u^ 3/ 3 + uv^2 , v − v^ 3/ 3 + u^ 2 v, u^2 − v^ 2 )$ is a minimal surface and x is not injective Proving that $x(u,v)$, ...
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28 views

Manifold arising from particular proof of Hairy Ball Theorem

Background, aka considerations to find my actual question In Geometry three, at the end of the last lesson, we sketched a proof of the famous Hairy Ball Theorem. The proof goes as follows. Lemma: ...
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1answer
40 views

Prove geodesics are straight lines if Riemann tensor is identically zero.

Suppose that $R^{a}_{bcd}\equiv 0$ in all of our manifold $M$ (in which we assume zero torsion). Prove that all geodesics are straight lines. I tried using Ricci's identity: ...
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1answer
23 views

Obstruction to the existence of constant-rank sections of $T^*M\odot T^*M$

If $\alpha$ is a section of $T^*M\odot T^*M$, where $M$ is a smooth manifold, the rank of $\alpha$ at $m\in M$ is the codimension of the kernel of $\alpha_m$, i.e. the subspace of vectors $v_m\in ...
2
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1answer
31 views

If a composition of two maps is smooth, as well as one of the maps, then so is the other.

Let $M$, $N$, and $K$ be smooth manifolds, and consider the maps $g:M\to N$, and $f:N\to L$. Assume that the composition $f\circ g$ is smooth. If any of $f$ and $g$ is smooth can we conclude that the ...
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22 views

The pullback bundle

Consider a smooth fibre bundle $P:E\to B$ over a base $B$ with fiber $F$, and a trivializing family $\{(\phi_a:P^{-1}(U_a)\tilde =U_a\times F\}_{a\in A}$. Let the clutch functions of this family be ...
4
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1answer
91 views

Is $\omega=\sin\varphi\,\mathrm{d}\theta\wedge\mathrm{d}\varphi$ an exact form?

Let $\omega=\sin\varphi\,\mathrm{d}\theta\wedge\mathrm{d}\varphi$ be a $2$-form on $\mathbb{R}^3\setminus\{0\}$. Then ...
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Construct a parametrization of a regular surface from a former parametrization such that is an isometry.

Let $\phi=\phi(u,v)$ be a coordinate neighborhood of a regular surface. And its coefficients from the first fundamental form $E,F,G$ are constants. Prove that there exists a change of parameteres ...
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1answer
69 views

An interpretation of $ \frac{\partial^{2}}{\partial x^{2}} $.

$ \left( \dfrac{\partial}{\partial x} \right)_{p} $ is both an element of the tangent space $ {T_{p}}(M) $ and a linear functional on $ {C^{1}}(M) $, while $ (\mathrm{d}{x})_{p} $ is an element of the ...
2
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1answer
55 views

Particular function in proof of flow box theorem

Flow Box Theorem If $M$ is a manifold of dimension $n$ and $X$ is a vector field on $M$ such that for a certain $p\in M$ $X(p)\neq0$, then there exists a chart $(U,\phi)$ on $M$ such that ...
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33 views

Basic question: Riemannian Curvature is nondegenerate

$R(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}$ is the curvature with respect to the Levi-Civita connection $\nabla$ of a metric $g$ on a manifold $M$. Define the Riemann curvature ...
2
votes
2answers
929 views

Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...
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73 views

Good video lectures in Differential Geometry

I was not fortunate enough to learn Differential Geometry during my Masters. As now I am having my thesis in PDEs, and I miss a lot of mathematics from the people who do PDEs on Manifold setting. I ...
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1answer
23 views

Prove that the normal to a quadratic curve passes through a specific point

I've been asked to prove that the normal to the curve $y=2x^2 - 3x^{-1/2}$ at the point $(1,-1)$ passes through the point $(12,3)$. $\frac{dy}{dx} = 4x + \frac{3}{2}x^{-3/2}$ Hence, at the point ...
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0answers
34 views

Flow of a left invariant vector field on a Lie group equipped with left-invariant metric and the group's geodesics

I think the answer to my question is known to many other people, but I'm still getting confused. Let $G$ be a (possibly infinite dimensional also) Lie group and $g$ be its Lie algebra. Consider the ...
2
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1answer
35 views

Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
2
votes
2answers
49 views

The difference between the algebraic torus and the geometric torus

I know that the donut-shaped geometric object in $\mathbb{R}^3$ is homeomorphic to a square with identified opposite sites. However, while the latter has a clear symmetry between two dimensions, the ...
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1answer
42 views

Covariant derivative (or connection) of and along a curve

Let $c:[a,b] \rightarrow M$ be a curve parametrized by arc-length. Then $c': [a,b] \rightarrow TM$ such that $c'(t) \in T_{c(t)}(M).$ So essentially, I want to understand how $\nabla_{c'}c'$ is ...
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1answer
54 views

Exercise 2.3 Lee's Riemmanian Manifolds

Statement: Suppose $M\subseteq \tilde{M}$ is an embedded submanifold. a)If $f$ is any smooth function on $M$, show that $f$ can be extended to a smooth function on $\tilde{M}$ whose restriction to ...
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1answer
77 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
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45 views

Example of orthogonal parametrization of a surface

I recently came to know about the orthogonal parametrization of a surface, for which $F={\bf X_u}\cdot{\bf X_v}=0$ and $E={\bf X_u}\cdot{\bf X_u}=G={\bf X_v}\cdot{\bf X_v}$. Here, $(E,F,G)$ denote the ...
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0answers
40 views

Is this differential 2-form closed

Consider a unit sphere $S^2 \subset R^3$ and a map $\omega_p : T_pS^2 \times T_pS^2 \to \mathbb{R}$ defined by $$\omega_p(u,v) = (u \times v) \cdot p$$ How do I know is this 2-form (on $S^2$) closed ...
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1answer
11 views

Express covariant transformation conveniently

Let $\omega = \sum_i \omega_i dx^i = \sum_{i} \nu_i dy^i$ be a 1-form in two different bases. Now, $(\omega_1,...,\omega_n)$ transform covariantly to $(\nu_1,...,\nu_n).$ My question is, can we ...
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1answer
28 views

Showing that an equation of a curve in the plane defines a surface in $R^3$.

A generalized cylinder is a ruled surface for which teh rulings are all Euclidean parallel. Thus there is always a parametrization of the form $$\mathbf{x}(u,v)=\beta (u)+v\mathbf{q} \; ...
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1answer
21 views

Poincaré lemma and conservative vector fields

Let $U$ be some contractible neighbourhood of $0\in\mathbb{R}^n$ and let $X=\sum_{i=1}^nX_i\frac{\partial}{\partial x_i}$ be a (smooth) vector field on $U$. This vector field can be thought as a ...
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1answer
35 views

Poincare: Change of form to a primitive 1-form

Given a 2-form w on $R^3$, such that dw=0; how does one find all 1-forms k such that dk=w? Provided a homotopy H(t) one can pull back w and apply the Poincare operator on it. But what if one does not ...
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1answer
20 views

Deduce that there is a cup product that is well-defined

I have showed that if $\alpha$ and $\beta$ are closed forms on a smooth manifold $M$, then $\alpha \wedge \beta$ also be closed. Further, if one of $\alpha$ or $\beta$ is exact, than $\alpha \wedge ...
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1answer
27 views

Smooth function from function with singularity

Having an application $f:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{C}$ in $C^{\infty}((t_0-\varepsilon, t_0+\varepsilon))$ with $f(t)=0\Leftrightarrow\ t=t_0$ and knowing that: $\exists\ ...
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1answer
26 views

Globally defined exponential in a particular homogeneous space

I'm currently working in a particular conformal compactification/completion of the Minkowski space-time, but I'm stuck at showing that the exponential of every vector field in it is globally defined. ...