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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Questions about compact orientable manifold, need your help!

$13.$ Suppose $M$ is compact orientable $n$-manifold (with no boundary), and $\theta$ is an $(n-1)$-form on $M$. Show that $d\theta$ is $0$ at some point. This is another problem in Spivak. ...
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conjugating a local flow by a diffeomorphism

Hi could someone please help me with the following problem: given F a diffeomorphism between M and N and f being a local flow of vector field v, i need to show that $FtF^{-1}$ is a local flow of ...
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Finite volume ends and Exponential decay

Assume that a noncompact manifold has finite volume and one of its ends is of the form $$\Sigma\times\left[0,\infty\right)$$ with $\Sigma$ compact and$\left[0,\infty\right)$ parametrized such that ...
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Flows and vector fields in differential geometry

given vector fields $f = f_{0} + L(x)$ and $g = g_{0} + G(x)$ of $\mathbb{R}^{n}$ where $L$ and $G$ linear, can someone help with finding the flow $\psi^{t}$ of $f$ and the bracket of $f,g$ Thanks ...
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Is a compact K-contact structure the same as compact toric contact structure of Reeb type?

I ran into this confusion when studying some physics problem. On a compact K-contact manifold, the Reeb vector field generates a 1-parameter family subgroup of isometries, and therefore its closure ...
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Getting Started with the Weil homomorphism

I have been trying to understand how invariant polynomials when applied to the curvature 2 form of a connection produce a differential form. I see how one obtains a globally defined tensor but not a ...
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Dual coordinates

If we have $R^{4}$ with basis $\{e_{1},e_{2},e_{3}=Je_{1},e_{4}=Je_{2}\}$. If we choose a surface $S=\{e_{1},e_{2}\}$ in $R^{4}$ and $x^{i}$ the coordinates dual to basis $e_{i}$. In order that S be ...
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Express the change of coordinate matrix in terms of partial derivative of the transition map $\phi$ Where $\tilde{\sigma} =\sigma \circ \phi$

Let $\sigma :U \to W\cap S$ and $\tilde{\sigma}: \tilde{U} \to \tilde W \cap S$ Be two surface patches around $p\in W\cap \tilde W$ $T_p(S) $ be tangent plane of S at p. $T_p(S)=span\{\sigma_u, ...
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53 views

chain rule for differential forms?

Let $f:U \to V$ be (smooth) map, where $U,V \subset \Bbb{R}^n$ are open subsets. Let $X(t)\in U$, then I saw an equation like $$\frac d {dt} f(X(t)) = df(X(t))X'(t)$$ But I'm not understanding what ...
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154 views

Convex curve in R^2 pass through two point with fixed normal direction

Given two distinct point in $\mathbb{R}^2$ and two distinct normal directions what are some explicit convex $C^2$ curve (e.g. a map $[0,1] \to \mathbb{R}^2$ which satisfy many equivalent property) ...
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61 views

find the estimated coordinate from a transformed system

Hello let me explain the situation We have the cartesian coordinate system in A. And then, it was transformed to become B. B as you can see is a transformed version of A. The red points signifies the ...
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67 views

Confusion on the meaning of “the opposite sides” in below problem.

Below is a problem which states a fact about "Tchebyshef net". I don't understand meaning of bolded part. Please help me. The coordinate curves of a parametrization $x(u, v)$ constitute a ...
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67 views

Inverse of a pythagorean hodograph curve.

I'm trying to solve this question about pythagorean hodograph curves: If $\alpha$ is a Pythagorean hodograph curve, and $\alpha(t)\neq(0,0)\;\forall t$, then the curve $\frac{1}{\alpha(t)}$ is also a ...
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30 views

arbitrary patch in terms of orthogonal patch

Let $\mathcal{M}$ be a $2$ dimensional differentiable manifold (a geometric surface). Let $\mathbf{y}$ be an arbitrary patch in $\mathcal{M}$. How can one prove that $y$ can be expressed as ...
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19 views

Existence of an integratiag factor (in the proof of isothermal parameters in analytic case)

Suppose $f(x,y),g(x,y)$ are analytic functions, does there exist a function $\lambda(x,y)$, such that $\lambda(fdx+gdy)=dh$ for some $h(x,y)$?(locally)
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area of a flat torus of revolution

Let $R>r>0$. We get a torus $T_0$ from the map $F(u,v)=((R+r\cos u)\cos v, (R+r\cos u)\sin v, r\sin u)$. Now define a new metric tensor on $T_0$ by $(x_u,x_u)=(x_v,x_v)=1, (x_u, x_v)=0$. ...
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49 views

Stuck on proving a surface is a cylinder or a plane

A surface with first and second fundamental forms $\mathrm{I}$ and $\mathrm{II}$ whose coefficients(i.e. $E,F,G,L,M,N$ with $\mathrm{I}=Edudu+2Fdudv+Gdvdv,\mathrm{II}=Ldudu+2Mdudv+Ndvdv$) are all ...
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114 views

Vector field and triangulation

I'm writing a paper on Poincaré-Hopf theorem about vector field indices and Euler Characteristic over topological compact surfaces and I got struck finding details on the last part of the proof. Let ...
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52 views

Curve on $S^2$ (2-sphere)

Let $\gamma$ be a curve on the 2-sphere $S^2$. Then: i) $k_n = |\gamma''|$ is constant. ii) $k_g=\langle t'(s),n(s)\rangle$ is constant. iii) $\gamma$ is a geodetic line, i.e. $k_g\equiv 0$ if ...
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42 views

Angle between unit vectors

I have a differential geometry problem... Given two unit vectors $X^r$ and $X^r+dX^r$ of the same curve, show that the angle between them is: $\theta^2=g_{mn}dX^mdX^n$. I understood how to do it with ...
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35 views

Definition of standard connection on $\mathbb{R}^n$ vs acceleration vector

I am currently looking at a set of lecture notes where the standard covariant derivative $D_XY$ in $\mathbb{R}^n$ is defined to motivate the more general concept of a connection. For two vector ...
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29 views

Existence of vector extensions for the Hessian

Q: $p$ a critical point of a smooth $f$ and $v, w$ two vectors in $T_{p}M$. We extend these two to vector fields $v^{*}, w^{*}$ such that at $p$ the first one equals $v$ and the second one equals $w.$ ...
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Differentiation along a curve on a manifold (Re: Schutz's intro to GR)

I am trying to show (Schutz chpt. 6 prob 13) that if two vector fields $\vec{A}$ and $\vec{B}$ are parallel transported along a curve $\gamma:\mathbb{R}\to M$ with real parameter $\lambda$ ($M$ a ...
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19 views

Continuous groups of Transformations [Reference request]

I am considering reading the book : 'Continuous Groups of Transformations' by Luther Pfahler Eisenhart. It seems to have a very interesting table of contents. However this is quite old and I am ...
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150 views

Tangent surfaces locally isometry on the plane

Let $\alpha : I \rightarrow \mathbb{R^3}$ be regular parametrized curve with the curvature $k(t) \neq 0$, $t\in I$. Let $$ \mathbb{X}(t,v) = \alpha(t) + v \alpha'(t) $$ be tangent surface. Prove ...
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189 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha ...
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A Survey on Rota's Conjecture

I am looking for a survey on Rota's Conjecture (1970). I am more interested in geometric aspects of the conjecture and any geometric content related to the conjecture. Any reference would be helpful. ...
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kernel of a smooth map

Let $M,N$ be smooth manifold , and let $f:M\rightarrow N$ be a smooth map, such that $y\in N$ is a regular value of it. Denote $M'=f^{-1}(y)$. Lemma: $ker(df_{x})=TM'_{x}$ for every $x\in M'$, ...
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Preimage of a regular value is manifold , for smooth map between smooth manifolds

I was reading the following Lemma: Let $M,N$ be manifold of $m\geq n$ dimensions respectively , and let $f:M\rightarrow N$ be smooth, then for every regular value $y\in M$ , the Preimage $f^{-1}(y)$ ...
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principal curves of a flat surface.

Let M be a surface in $R^3$ with principal curvatures $k_1=0, k_2\neq 0$. $k_1$ is always zero and $k_2$ is never zero. Suppose that $E_1, E_2$ are corresponding principal unit vector fields and ...
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conformally flat 3-manifolds

Does there exist closed three manifolds that don't admit a metric that is locally conformally flat? Recall that a three manifold is locally conformally flat if the exterior derivative of the Schouten ...
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Tangent space on a differentiable function

I am wondering about a definition that I found in my notes: Let $f$ be a differentiable function at a point $x_0$ and $Df(x_0)$ denote its derivative, then $\text{Graph(Df(x_0))}:=\{(h,Df(x_0)h)\}$ ...
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Observation on normalized Ricci flow on two sphere

Note that two sphere with nonnegative curvature converges to canonical sphere along normalized Ricci flow. So assume that if curvature is bounded below by $-\epsilon$ then the sphere meets a ...
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33 views

An abstract definition of the cotangent space to a smooth manifold.

I need a book introducing the cotangent space to smooth manifolds in the most abstract way. So $T^\ast_p M$ by this point of view should be the quotient ring $I/I^2$ where $I=\{[f]\in ...
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Normal curvature of a circle in a plane

I have the circle $\gamma(t) = (\cos t, \sin t, 0)$ in the plane $z=0$. Now I understand that normal curvature is related to the second fundamental form, and an expression for it is ...
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lie group homomorphism and compactness

Let $f\colon M\rightarrow N$ be a surjective homomorphism of lie groups. Assume the follwoing: f has discrete kernel $ker(f)=\{e,-e\}$ a path from $e$ to $-e$ exisits $N$ is connected Claim: $M$ ...
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Manifolds : Show that a maximal $C^r$-atlas containing a $C^r$-atlas $A$ is unique.

So I want to show that a maximal $C^r$-atlas containing a $C^r$-atlas $A$ is unique. I believe I have shown this, but in my proof I require the fact that the charts of $A$ cover our manifold in ...
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Volume and Ricci curvature

If the Ricci curvature is zero on a manifold, does that mean that, if I choose a sphere and let each point contained in that sphere move along geodesics that are all initially parallel, the volume of ...
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172 views

total Gaussian curvature of a surface

I have to evaluate a total Gaussian curvature of the surface in $\mathbb{R}^3$ : $x^2 + y^2 - z^2 = 1$ I tried to solve the problem by using Gauss-Bonnet Theorem or direct computation. When I used ...
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Chain of transformations -> continuous

Transformation $A_t$ rotates point $p(t)$ for angle $d\phi(t)$ around the axis $n(t)$ anchored at point $r(t)$ and finally displaces it for $r'(t) dt$. Point is now $p(t + dt)$. More specifically in ...
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French translation, and what is the curvature of a metric?

I have a french paper to read. There is the notion of une collection des courbures des métriques $g_t$. Now I would guess that this refers to a collection of curvatures of metrics $g_t$, however ...
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Vector bundle addition

Let $\pi:E \to M$ a smooth vector bundle and let $E \times _M E= \{(x,y) \in E \times E| \pi(x)=\pi(y) \}$. I want to show the addition map $E \times_M E \to M, (x,y) \to x+y$ is smooth. The ...
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Examples of totally real minimal submanifolds

If we have $R^{4}$ with basis $\{e_{1},e_{2},e_{3}=Je_{1},e_{4}=Je_{2}\}$, then we know that $\{e_{1},e_{2}\}$ is totally real minimal submanifold of $R^{4}$. Is there a nontrivial example of totally ...
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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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60 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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66 views

how to determine the curve

I have to determined the curve which passes through the point $(1/2, \sqrt3 /2)$ and cuts to each member of the family of circles $x^2+y^2=a^2$ forming a angle of $60º$ My idea is to create a ...
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51 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 ...
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group of automorphism of an $SU(2)$-bundle

Let $H$ be the Hopf bundle over $S^2$, consider the vector bundle $H^k \bigoplus H^{-k}$ over $S^2$ and extend it to $S^2 \times R^+$, this gives an $SU(2)$-bundle $E_k$. The claim is that the group ...
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83 views

Dupin's indicatrix and asymptotic direction of a surface?

Asymptotic direction at a point $p$ of a surface $S$ is defined to be the direction of $T_{p}(S)$ for which the normal curvature is zero. And Dupin's indicatrix at a point $p$ of surface $S$ is ...
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Sections on the Tautological Line bundle $E(\gamma_n)$..

I have a question about the tautological line bundle over $\mathbb R\mathbb P^n$. Recall, this bundle is that whose total space is $$E(\gamma_n):=\{([x], v)\in\mathbb R\mathbb P^n\times \mathbb ...