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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How would i find the volume of a cone in the $interval [0,a]\times[0,a]\times[0,a]$ and how it's surface area? (using integration?)

whichEssentially i want to find the measure of $z^2\leq x^2+y^2$ and $z^2=x^2+y^2$. Now i know for one of them i would incorporate cilindrical coordinates: $$g(r,\phi,z)=(rcos \phi, r sin\phi,z)$$ ...
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35 views

Orthonormal frame on hyperbolic plane

I'm having trouble comprehending a question from Do Carmo's Differential Forms and Applications. The question (in its entirety) is as follows: (Exercise 5-2 in Do Carmo). Let $H^2$ be the upper ...
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Zero Gauss curvature and constant mean curvature of a ruled surface in $\mathbb{R^3}$ implies it is a right cylinder

Assuming I have a ruled surface parametrized as $x(u,v)=\beta(u)+v\delta(u)$, with zero Gauss curvature, which in this case is given by $K=-\frac{m^2}{EG-F^2}$=$\frac{- (\beta'\delta \times \delta')^2 ...
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171 views

Why would one care about Fibre Bundles

As a physics student I can easily understand the motivation for studying manifolds and why the definition looks the way it does, I only have to think of Minkowski space in GR. But for the life of me ...
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23 views

Let $G=GL(n,\mathbb R)$, show that this application $ (A,B) \in G \times G \rightarrow AB \in G$ is $C^{\infty}$

Let be $G=GL(n,\mathbb R)$. I consider the application $$a: G \times G \rightarrow G$$ such that $$ (A,B) \rightarrow AB .$$ I have to prove that this application is $C^\infty$. I know the ...
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1answer
38 views

Induced Connection on $\Sigma\subset M$

Let $(M,g)$ be a Riemannian manifold, $\Sigma$ a manifold and $F:\Sigma \rightarrow M$ a smooth map. For $X,Y \in \Gamma(T\Sigma)$ vector fields and $\tilde{\nabla}$ the pull back connection on ...
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226 views

Metric on Steifel and Grassmannian manifolds generalizing Fubini-Study

If $F$ is $\mathbb{R}, \mathbb{C}$, or $ \mathbb{H}$, the Grassmannian manifold $G_k(\textbf F^n)$ is the space of all $k$ dimensional subspaces of the $n$ dimensional vector space $F^n$. The Stiefel ...
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1answer
71 views

(co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...
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1answer
32 views

Let $\beta$ be a unit speed curve with $\kappa \gt 0$. Show that $(\beta''\times \beta''')\cdot \beta^{4}=\kappa^5\frac{d}{ds}(\frac{\tau}{\kappa})$

Let $\beta$ be a unit speed curve with $\kappa \gt 0$. Show that $$(\beta''\times \beta''')\cdot \beta^{4}=\kappa^5\frac{d}{ds}(\frac{\tau}{\kappa})$$ Simple calculation seems too frustrating. I'm ...
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47 views

A compact $n-1$ dimesional manifold embedded in $\mathbb{R}^n$ has no boundary

Under what conditions is it true that a compact $n-1$ dimesional manifold embedded in $\mathbb{R}^n$ has no boundary (or more generally, when a manifold is embedded in some topological space)? For ...
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1answer
64 views

If $ g \circ f$ is real analytic and $g$ is a real analytic immersion, then $f$ is real analytic

Let $M$ $N$ $P$ be complex manifolds, and let $$f: M\rightarrow N, g: N\rightarrow P$$ be $C^\infty$ maps with $g$ and $g\circ f$ holomorphic, and with $dg$ never degenerate. It's easy, then, to see ...
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1answer
31 views

Projected Area of Circle onto the Side of a Cylinder

I have encountered a problem that requires me to find the projected area of a beam of light (circular cross-section, with radius R1) vertically onto the side of a cylinder with R2 as its ...
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40 views

If $\alpha$ is a unit speed curve of constant curvature lying in a sphere, then $\alpha$ is a circle.

I'm trying to solve the following problem but got stuck along the way. I would like some help on getting this through. Prove that if $\alpha$ is a unit speed curve of constant curvature lying in a ...
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1answer
45 views

Problem finding regular values of map

I found an exercise defining $f:S^3\to\mathbb{CP}^1$ by $f(x,y,z,t)=[x+iy:z+it]$ and asking to prove it was smooth and find its regular values. Proving it was smooth was simple enough. Then I tried ...
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1answer
63 views

Curvature in $\mathbb{R}^2$

Let $f(t) = (x(t),y(t))$, not necessarily parametrized by arclength. We define the unit tangent vector, $T(t) = (1/|f'(t)|)(x',y')$. Also the normal vector, $N(t) = (1/|f'(t)|)(-y',x')$, which is ...
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68 views

fiber bundle in topological category and smooth category.

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundle on $M$ with structural group $G$ in smooth category. And denote by ...
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1answer
16 views

Space curve torsion

Hello I am looking for anyone to maybe look over my ideas and see if they think it is correct. Say I am looking for the torsion $\tau$ of a space curve given by $r(t)=(cos(3t),sin(3t),4t)$ I know if ...
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2answers
527 views

Smooth maps (between manifolds) are continuous (comment in Barrett O'Neill's textbook)

(Needless to say, I'm a total newbie in differential geometry so I apologize if this seems rather too obvious to many of you). As a comment on his definition of smooth mapping, Barrett O'Neill in his ...
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139 views

Is there geometric interpretation to Skew symmetric coefficient matrix,

We know that the Frenet-Serret equation implies that the coefficient matrix of $\dot t,\dot n,\dot b$ is anti symmetric wrt $t,n,b$. But is there any geometric intuition that immediately gives this ...
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28 views

Cohomology of two pieces of torus

In an exercise from an old exam, I found myself confronted with $M=\{(\sqrt{x^2+y^2}-2)^2+z^2=1\}$, $U=M\cap\{x\neq0\vee y>0\}$, $V=M\cap\{x\neq0\vee y<0\}$ and $U\cap V$, all subsets of ...
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69 views

Killing vector field along a geodesic

I was trying to show that a Killing vector field satisfies the Jacobi Equation for a geodesic, just by assuming that \begin{equation} \nabla_\mu X_\nu + \nabla_\nu X_\mu=0 \end{equation} Indeed, if I ...
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59 views

Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$

I'm lost on solving the following problem. Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$, where $\theta$ is the angle between the positive $x$-axis and the ...
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1answer
65 views

Metric on n-sphere in terms of stereographic projection coordinates

The metric on the $n$-sphere is the metric induced from the ambient Euclidean metric. Find the metric, $d\Omega^2_n$, on the $n$-sphere and the volume form, $\Omega_{S_n}$ , of $S^n$ in terms of the ...
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1answer
34 views

How to show that two vector fields commute?

Could anyone help me with how to start to solve the following problem? From this problem as well as this, I have: Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such ...
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18 views

Find $d\left(\frac{\partial\left(x,y\right)}{\partial\left(\delta_1,\delta_2\right)}\right)$ with the exterior product

Let $J_{\delta_1,\delta_2}^{x,y}$ denote the Jacobian $\partial\left(x,y\right)/\partial\left(\delta_1,\delta_2\right)$. Suppose I wanted to find $d\left(J_{\delta_1,\delta_2}^{x,y}\right)$ ...
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1answer
33 views

A diffeomorphism whose tangent map preserves dot products is an isometry.

I'm having trouble solving the following problem. If $F:\mathbb{R^3} \to \mathbb{R^3}$ is a diffeomorphism such that $F\ast$(the tangent map of $F$) preserves dot products, show that $F$ is an ...
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60 views

$\psi ( \frac {\pi}{2}, \frac {\pi}{6})$ and calculating problems? [closed]

I ran into a problem, $u=\psi (x,t)$ be a solution of partial deferential equation with following condition on boundary, how we reach the value of $\psi ( \frac {\pi}{2}, \frac {\pi}{6})$? ...
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48 views

Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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21 views

Proof that any unit-speed-reparametrization of a curve preserves orientation and is an inverse of an arc length function based at some $t_0$.

I am not able to prove the following two facts about a unit speed reparametrization of a curve. Let $\alpha$ be defined on some interval $I$ and define for $t_0\in I$ ...
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28 views

An orthogonal transformation with determinant 1 rotate $\mathbb{R^3}$ around an axis.

I'm studying differential geometry and need confirmation on the solution of the following problem. A rotation is an orthogonal transformation $C$ such that det $C=+1$. Prove that $C$ does, in fact, ...
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1answer
45 views

How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?

Could anyone help me with the following problem? The problem Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for ...
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3answers
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is every totally geodesic submanifold the set of fixed points of some isometries?

It is well known that the set of fixed points of an isometry $\phi:(M,g)\rightarrow (M,g)$ is a totally geodesic embedded submanifold. (e.g here ). I ask whether the converse is true, i.e is every ...
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1answer
42 views

immersion of punctured torus in plane

Let $S^1 \times S^1$ be the $2$-torus. If a point $a=(p,q)$ of the torus is removed, i.e., it is punctured at one point then how can I show that it can be immersed in the plane, i.e., in $\Bbb{R}^2$? ...
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333 views

Unit speed reparametrization of curve

I am learning Elementary Differential Geometry by O'Neill and having a hard time with this exercise. Suppose that $\beta_1$ and $\beta_2$ are unit-speed reparametrizations of the same curve $\alpha$. ...
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Associated bundle - valued 1 forms.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Fact: (as can be found in lecture notes here) Functions ...
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Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
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1answer
24 views

Jets and vertical differential

For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $x\in M$ and $u=\phi (x)$. The vertical differential of the section $\phi$ at point $u\in E$ is the map: ...
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Vector fields that are smooth on the open unit ball $B$, are smooth on $\mathbb{R}^n$ if they are zero outside $B$

Could anyone help a little with the following problem? It is a continuation of this problem, but I will restate the things that are needed: Fix $\varepsilon \in (0, 1)$ and choose a smooth ...
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3answers
41 views

Derivative of a function in $\mathbb{R}^n$

Let $f:\mathbb{R}^m\to\mathbb{R}^m$ be a differential function. Let $Df(x)$ be the derivative of f at $x\in\mathbb{R}^m$. Which of the following is/are correct? $Df(0)(u)=0 \forall u\in\mathbb{R}^m$ ...
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Existence of a differentiable function given a unit gradient field

I'm trying to prove that "Given a unit vector field $V$, it can always be uniquely determined a differentiable function $f$ that satisfies $\nabla f = V$." To provide you more information, the unit ...
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756 views

Spivak and Invariance of Domain

On p.3 of the first volume of Spivak's Comprehensive Introduction to Differential Geometry, he says that it is an "easy exercise" to show that the invariance of domain theorem (if ...
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464 views

Equation of a straight line in spherical coordinates

I'm trying to prove the angle sum formula for a triangle on the surface of a sphere. In order to do this I wanted to create a general triangle on the sphere, with one vertex at $\theta = 0$ and one ...
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1answer
24 views

Basic properties of smooth curves

Suppose $\Gamma$ is simple smooth closed curve parametrized by $\gamma:[0,1]\to\Gamma.$ Let $$\gamma(t)-\gamma(s)=(t-s)F(t,s)\,\,\,\,t,s\in[0,1].$$ Can we conclude that $|F(t,s)|>0$ for all ...
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1answer
107 views

Transversality of Vector Fields Defined in terms of Diff. Forms and Open Books.

All: Sorry for the length of the post, but I think it is necessary to set things up so that the post is understandable. I'm trying to understand how it is that the transversality (in this case , the ...
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What is some elementary differential geometry textbook that is self contained and are intermediate level?

What are some intermediate differential geometry textbook that are more advanced than pressley's, Barrett's, Christian's and krezig's books and are self contained but below the level spivak's vol ...
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1answer
73 views

Proving a certain function is injective

I have found the following exercise on an exam for Geometry three dating to a past year. Let $F(u,v)=((2-v\sin\frac{u}{2})\sin u,(2-v\sin\frac{u}{2})\cos u,v\cos\frac{u}{2})$, with ...
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106 views

The meaning of variables and derivations in Souriau's book

As far as I see, Souriau is using unconventional notions in his book "Structure of Dynamical Systems". He explains these notions in §2. of Chapter I, but it is a puzzle for me. Mainly because he ...
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1answer
38 views

How to proof that bracket of two vector field can be computed by second derivation

Can some one give a hint how can I proof that where $\phi$ indicated the flow of vector fields.
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28 views

Green's theorem via Stokes's theorem

I am considering the following form of Stokes's theorem: Let $\omega$ be an $n-1$ differential form with compact support on an oriented manifold of dimension $n$. Let us consider the boundary ...
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34 views

Pullback 1-differential form

Let $(x_0,x'_0) \in \mathbb{R}^2$ be initial data for the Euler Lagrange equation with some given Lagrangian $L: \mathbb{R}^2 \rightarrow \mathbb{R}.$ Then $F^t$ is the flow that maps the initial ...