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 ...

learn more… | top users | synonyms (1)

1
vote
0answers
7 views

Description of $(T\Bbb{CP}^1)^\perp$

Is there a nice "concrete" description (i.e., coordinates) of the normal bundle of $\Bbb{CP}^1$ when is considered as a submanifold of $\Bbb{CP}^n$? Or, at least, $\Bbb{CP}^2$?
2
votes
0answers
7 views

Fibre bundles for homogeneous spaces

Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. Then it is known that $G/H$ can be equipped with a unique differentiable structure such that $G\xrightarrow{\pi} G/H$ is a locally ...
1
vote
1answer
19 views

Composition of smooth maps between manifolds is smooth

This is a continuation of the problem : Composition of smooth maps. At the moment, I am on the same problem. I am not quite sure of the continuation of the comment '' The point here is another. Are ...
1
vote
0answers
11 views

Index of a zero of a normal vector field

Let $M$ be a manifold and $S\subset M$ a submanifold. Assume whatever regularity you need (smoothness, compactenss, orientability, empty boundary, proper embedding of $S$,... this question is ...
1
vote
0answers
16 views

How to define the boundary operator using the exterior derivative?

I am looking for a way to define the boundary operator $\partial : M^n \to N^{n-1}$ from an $n$-dimensional manifold $M$ to its boundary $N$ using the the expression \begin{equation*} \int_M d \alpha ...
2
votes
1answer
16 views

A regular surface with non zero mean curvature is orientable

How can I prove that any regular surface with non zero mean curvature is orientable?
2
votes
0answers
26 views

Counterexample for Sard's Theorem Replacing “Smooth” with “Differentiable”

Is there a differentiable (but not continuously differentiable) function f between Euclidean spaces (or manifolds, whatever) such that the set of critical values of f has nonzero measure?
1
vote
1answer
12 views

Preimage surfaces.

Let $U\subset\mathbb{R}^{m+n}$ open set, $f:U\longrightarrow\mathbb{R}^n$, $f\in C^{k}$, and $c\in\mathbb{R}^n$. Set: $$M=\{p\in U; f(p)=c\textrm{ and ...
2
votes
0answers
33 views

Div, grad, curl in curvilinear coordinates

I've a lot of different formulas for div, grad, curl, and laplacian in different coordinate systems. How are these formulas derived? What's the general procedure for finding the formula of say the ...
0
votes
1answer
19 views

Finding the complex structure which induces an almost complex structure (example)

Given an almost complex structure (i.e. an endomorphism $\widetilde{A} : TM \rightarrow TM$ such that $\widetilde{A}^{2}=-$Id) defined by: $\widetilde{A}\left(\frac{\partial}{\partial x_{1}}\right) = ...
3
votes
0answers
21 views

Vector space operations on fibres of associated bundles.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\text{ad}:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let ...
1
vote
1answer
38 views

Show this is not a manifold with boundary

Consider a curve $\alpha: \mathbb R \to \mathbb R^2$ defined by $t \mapsto (e^t \cos(t), e^t \sin(t))$. Show the closure of $\alpha(\mathbb R )$ is not a manifold with boundary. Denote ...
2
votes
1answer
39 views

Integration over ellipse

$A=\{(x,y)\in \Bbb R^2\mid \frac{x^2}{a^2}+\frac {y^2}{b^2}=1\}$. Find $\int_A (\cos x)y\,dx+(x+\sin x)\,dy$. Can someone please please give a methodological answer? Thanks a lot!
1
vote
2answers
49 views

Why spheres are not symplectic manifolds?

Reading some books on diferential geometry, a found that $S^{2n}$ (with $ n>1$) are not symplectic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not ...
0
votes
0answers
39 views

Dirac delta question from “Classical covariant fields” by Burgess

If you have the book with you. Kindly tell me how did he reach equation 2.54 from equation 2.52. I tried to solve the delta function according to given instruction but I am making some mistake. Kindly ...
1
vote
1answer
24 views

For a positive definite quadratic form $f: R^n \rightarrow R$, $f^{-1}(x)$, for any $x>0$, is diffeomorphic to $S^{n-1 }$

How to show for a positive definite quadratic form $f: R^n \rightarrow R$, there exists $f^{-1}(x)$, for any $x>0$, is diffeomorphic to $S^{n-1 }$?
1
vote
1answer
19 views

Obvious way to $F$-relate vector fields?

An exercise in Lee's Introduction to Smooth Manifolds asks one to check that $F:\mathbb R\to\mathbb R^2$ given by $F(t)=(\cos t,\sin t)$ relates $X=d/dt\in\mathfrak X(\mathbb R)$ to $Y\in\mathfrak ...
5
votes
1answer
34 views

locus of moving circles with changing radius

Suppose I a curve, $\gamma(t) = (x(t),y(t),R(t))$, which describes the centroids, $(x(t),y(t))$ and radii, $R(t)$, of an infinite number of circles parameterised by $t\in(a,b)$. I would like to find ...
0
votes
0answers
46 views

Integrating 2-form over torus

Let $\Bbb M^2 ⊂ \Bbb R^3$ be the torus of revolution obtained by rotating the circle $(x−2)^2 +z^2 = 1$ in the $xz$ plane around the $z$ axis. Consider the orientation on $M$ induced by the ...
0
votes
1answer
24 views

Unsolved regular value problem in $\mathbb{R}^n$

I want to show that if $F : \mathbb{R}^n \rightarrow \mathbb{R}^{n-k}$ is a $C^1$ function and $rank(DF) = n-k$ then $M:=F^{-1}(\{0\})$ defines a manifold. My idea: Without loss of generality I ...
1
vote
0answers
13 views

Average viewing angle of a convex body from a curve.

This is an integral geometry question. Let $K$ be a convex body in the plane, and $\mathcal{C}$ a simple closed curve the interior of which contains $K$. From each point $P$ of $\mathcal{C}$ two ...
1
vote
0answers
50 views

What's the difference between $Df$ and $Tf$?

I'm reading Michael Shub's Global Stability of Dynamical Systems. In chapter 4, he defined hyperbolic set and said the splitting $E^s$ and $E^u$ are $Tf$ invariant. So I assume this $Tf$ is the ...
0
votes
0answers
32 views

what is smooth embedding

I know definition but I don't use this definiton to solve question for example; $f : R → R^3$ given by $f(t) = (cos(2πt),sin(2πt), t)$ I showed that this is an injective immersion but this is a ...
1
vote
1answer
22 views

Prove that this application $f:S^n\rightarrow \mathbb{RP}^n$ is a local diffeomorphism, alternative approach using curves

I consider $f:S^n\rightarrow \mathbb{RP}^n$, the restriction to $n$-sphere $S^n$ of the canonical projection $\pi :\mathbb{R}^{n+1}\setminus \{0\} \rightarrow\mathbb{RP}^n $. I have to prove that $f$ ...
0
votes
0answers
24 views

Relationship between Cartan and Fréchet derivative

Let $f: X \rightarrow \mathbb{R}$ be smooth, then the Fréchet derivative is a map $Df: X \rightarrow L(X, \mathbb{R}).$ But if $f: M \rightarrow \mathbb{R}$ is smooth and $M$ a manifold, then the ...
1
vote
1answer
21 views

showing pushfarward

Let $M,N$ be two differentiable manifolds and $f:M \rightarrow N$ be a smooth map. Define a new map $F:M\rightarrow M\times N$ by $F(p)=(p,f(p))$ I can prove first part which is F is smooth but I can ...
3
votes
1answer
53 views

Open Unit Ball diffeomorphic to the Open Unit Cube

How can I show that the open unit cube $(-1,1)^n \subset \mathbb{R}^n$ and the open unit ball $B = \{x \in \mathbb{R}^n \mid \|x\| < 1\}$ are diffeomorphic? I know that one can proof this by ...
1
vote
1answer
24 views

Given $f : P\rightarrow N$ $C^\infty$ and $\pi : M\rightarrow N$ local diffeomorphism show that $\tilde f$ s.t.$f= \tilde f \circ \pi$ is $C^\infty$

Let be $M$, $N$ and $P$ three differentiable manifolds. I consider $\pi: M \rightarrow N$ a local diffeomorphism and $f:P \rightarrow N$ differentiable. I have to prove that the application $$ ...
2
votes
0answers
19 views

Reparametrization with non-vanishing lateral derivatives

Let $\mathbf{r}:I\subseteq\mathbb{R}\to\mathbb{R}^2$ be a $C^{1}(I)$ function, non-constant on any subinterval, such that $\forall\ t_0\in I$, the following limits exist: $\lim\limits_{t\nearrow ...
2
votes
1answer
38 views

Some detail in the proof of the mean value formulae for harmonic functions

Let $B_r(x_0)$ and $\overline{B}_r(x_0)$ be the open and closed ball in $\mathbb{R}^n$, respectively $u\in C^2(B_r(x_0))$ and $\rho\in (0,r)$ $\lambda_n$ be the Lebesgue-measure on the ...
1
vote
0answers
20 views

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: $$g(r,\phi,z)=(rcos \phi, r sin\phi,z)$$ ...
5
votes
2answers
168 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 ...
2
votes
1answer
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 ...
1
vote
1answer
30 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 ...
2
votes
0answers
45 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 ...
3
votes
1answer
33 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 ...
2
votes
1answer
28 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 ...
1
vote
1answer
60 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 ...
0
votes
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 ...
1
vote
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 ...
3
votes
0answers
39 views

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 ...
1
vote
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 ...
5
votes
1answer
64 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 ...
4
votes
2answers
29 views

basic question about holonomy

I'm struggling to understand how conditions on the metric put conditions on the holonomy group and vice-versa. My understanding is that the holonomy principle says that there's a one-to-one ...
1
vote
1answer
31 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 ...
1
vote
0answers
27 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 ...
0
votes
0answers
17 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 $d\left(J_{\delta_1,\delta_2}^{x,y}\right)$ ...
4
votes
1answer
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 ...
3
votes
3answers
54 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 ...
4
votes
2answers
52 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 ...