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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Working with homomorphisms and de Rham cohomology.

Here’s my question: Let M be a connected, compact, orientable, smooth n-manifold ($ n \in \mathbb{N}_{\geq 2} $). Let V be a neighborhood of p diffeomorphic to $\mathbb{R}^n$ and let U = M \ {p}. ...
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19 views

Intuitive interpretation of Ricci Flow

What is the best way to interpret, explain or somehow visualize the basic idea behind formal definition of Ricci Flow? I am familiar with the hackneyed expressions like "Ricci Flow is a ...
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13 views

Orthogonal surfaces

Prove that the three surfaces of the family $xy/z=u$ $\sqrt{x^2+y^2}+\sqrt{y^2+z^2}=v$, $\sqrt{x^2+y^2}-\sqrt{y^2+z^2}=w$ that pass through just one point are orthogonal I´m assuming that first I ...
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11 views

Holomorphic Hermitian metrics

Let $E\to M$ be a complex vector bundle. A hermitian metric $h$ on $E$ is a hermitian inner product on each fiber $E_{p},\, p\in M$. Suppose that $M$ is also a complex manifold and that $E$ is ...
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15 views

Properties of $\Omega_{\epsilon}$

Let $\Omega$ be a bounded connected domain in $\mathbb R^n$ with compact smooth boundary. So $\partial \Omega$ can be viewed as a smooth submanifold of dimension n-1. Let $$\Omega_{\epsilon} : = \{ ...
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Is Gauss curvature a Morse function?

Given a Gauss map $\nu: M \rightarrow S^k$ of a orientable, compact manifold, we define the shape operator $S_p = -d \nu: T_p M \rightarrow T_{\nu(p)} S^k$ to be the negative differential. Define the ...
2
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41 views

Why is it called the cotangent bundle?

We all know that the cotangent of an angle is the tangent of the complement of that angle. What is the etymology of a cotangent bundle? In the sense of mechanics, the coordinates of the tangent bundle ...
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18 views

Unknown functions yield a given determinant

I am trying to develop a nomogram which simultaneously shows the exact Fisher equation $(1+u) = (1+v)(1+w)$ and its linear approximation $u \approx v + w$. This amounts to finding twelve smooth ...
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14 views

Dependence among parameters for a geodesic [on hold]

A surface in 3-space is defined by: $$ x = X(u,v), y = Y(u,v) , z = Z(u,v) $$ Find differential relation or function connecting like $ v =f(u)$ or $ g( u,v, \frac{du}{dv}) =0 $ such that a ...
5
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2answers
70 views

Finding the de Rham cohomology of an open subset of $ \Bbb{R}^{n} $ minus a point.

Here’s my question: Let $ n \in \mathbb{N}_{\geq 2} $. Suppose that $ U \subseteq \Bbb{R}^{n} $ is an open set and that $ x \in U $. Then show that $$ {H_{\text{dR}}^{n - 1}}(U \setminus \{ x ...
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19 views

Smooth manifold

$A=M\cap N$, $M={(x,y,z\in\Bbb R^3)| x^2+y^2=1}$, $N=(x,y,z)\in \Bbb R^3|x^2-xy+y^2-z=1$. $1$. Is $A$ is smooth manifold? $2$. Find the points of $A$ that are farthest from the origin. This is ...
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22 views

Tangent space chart dependent or not?

I was wondering whether the definition of a tangent space is chart dependent or not? Cause to me it seems that they depend on the charts: Let $p \in M$ where $M$ is a k-dim smooth manifold and $\phi: ...
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0answers
35 views

Shifting to polar coordinates

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2,\ a,b\in\mathbb{R}, a<b$, be a $C^{\infty}([a,b])$ application (smooth). Is it true that we can always find two functions $r,\varphi :[a,b]\to\mathbb{R}$, $r\in ...
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12 views

Computing the index around a curve with respect to a field, invariance?

If I understood the course book Nonlinear Dynamics and Chaos right, The index can be found by $$\newcommand{\dd}{\mathrm{d}} \newcommand{\id}{\mathrm{d\,}} I_{C}=\frac{1}{2\pi}\oint_C ...
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51 views
+150

Hausdorff measure and volume form

I would like to find a proof of the following fact: If $M$ is an orientable $k$-submanifold in $\mathbb{R}^n$ with a volume form $dV$ then $$\int\limits_{M} f(x) dV = \int\limits_{M} f(x) H^k(dx).$$ ...
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31 views

How to show that each tangent vector to $\mathbb{R}^n$ at a point $a$ is of the form $\xi(f) = \sum_{i} c_i \frac{\partial f}{\partial x_i}(a)$? [on hold]

How to show that each tangent vector to $\mathbb{R}^n$ at a point $a = (a_1, \ldots, a_n) $ is of the form $\xi(f) = \sum_{i=1}^n c_i \frac{\partial f}{\partial x_i}(a)$? Thank you very much. Edit: ...
2
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1answer
34 views

a question about differential geometry(Gauss-bonnet theorem and isolated singular point in the surface)

Let C be a regular closed simple curve on a sphere $S^2$. Let v be a differentiable vector field on $S^2$ such that the trajectories of v are never tangent to C. prove that each of the two regions ...
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1answer
36 views

a question about undergraduate-level differential geometry(Gauss-Bonnet theorem)

Let $S\subset R^3$ be a regular surface homeomorphic to a sphere. Let $\alpha\subset S $ be a simple closed geodesic in S,let A and B be a regions of S which have $\alpha$ as a common boundary. Let ...
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1answer
33 views

Tangent bundle to a simple manifold.

Let $\mathbb{R}$ be the manifold of interest. So $\mathcal{M}$ = $\mathbb{R}$. We define a coordinate $x$ which gives us a point on the manifold. The tangent plane to $\mathcal{M}$ at a point $x=p$, ...
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1answer
47 views

Using stokes' theorem

$B=\{(x,y), x^2+y^2\le1\} $ is a closed ball and $S=\{(x,y,z), z=x^2+y^2, (x,y)\in B\} $ oriented so that $f:B\to S$ defined by $$f(x,y)=(x,y,x^2+y^2)$$ is orientation preserving. Compute ...
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28 views

Unique solution of a simple functional equation

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in ...
3
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1answer
58 views
+100

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...
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1answer
19 views

first Chern class of E is first Chern class of det E

Let $\pi:E\to M$ be a vector bundle, and $\nabla$ a connection. My definition of the first Chern class is $$c_1(E)=\left[tr\left(\frac{i}{2\pi}F^\nabla\right)\right],$$ where $F^\nabla$ is the ...
0
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1answer
26 views

how to find points where a k-form is nonvanishing.

for example, if we are given 2-form $\omega=2xdx\wedge dy+2ydy\wedge dz$, what are the points where the form vanishes? I can only think of points $(0,0,z)$, is it all? Additionally, if we have a form ...
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0answers
61 views

Integration over a manifold with boundary (Check).

Assume that $ f: \Bbb{R}^{3} \to \Bbb{R} $ is a smooth function such that $ M \stackrel{\text{df}}{=} \left\{ \mathbf{x} \in \Bbb{R}^{3} ~ \middle| ~ f(\mathbf{x}) \ge 0 \right\} $ is a non-empty ...
3
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1answer
58 views

Is it possible to compute geodesic without induced metric

Suppose a manifold embedding $i:M\to N$ into Riemannian manifold $(N,g)$ is given by $f(x)=0$, where $f:M\to R^m$ is a smooth vector-valued function. Now if it is very hard to parameterize the ...
2
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1answer
80 views

Why klein Bottle is 4-D?

I am wondering that Klein Bottle is 4-D. Can any body tell me how it is possible? I can give coordinates for each point of the Klein Bottle with 3 values. Then how it can be 4-D? What is immersion? ...
3
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1answer
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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 ...
2
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1answer
32 views

An Application of Stokes's Theorem

Let $D^2=\{(x,y)\in \mathbf R^2: x^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^2$, and $D^3=\{(x,y,z)\in \mathbf R^3: x^2+y^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^3$. Let $i_{\pm}:D^2\to ...
5
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3answers
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Parametrization of the lemniscate

All over the net, it is stated that the parametrization of the lemniscate with Cartesian equation $(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)$ is: $$\varphi: t \mapsto ...
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23 views

Sp(2n) as manifold

How to prove that $Sp(2n)$ is a manifold? We know that $Sp(2n)\subset Gl(2n)$ and $Gl(2n)$ is a manifold. Furthermore $Sp(2n)$ can be described as zeros of $A\mapsto A^TJA-J $, where $J$ is a ...
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36 views

Stokes Theorem on a sphere problem

I'm looking through my multivariable calculus notes and have come across a question I'm not sure I fully understand. It reads, "If $\omega$ is a differential form on $\mathbb{R}^3$ and $M$ is a sphere ...
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1answer
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Is the form closed?

$S$ is an n dimensional unit sphere such that $S^n=(x\in \Bbb R^{n+1}: |x|=1)$ with some fixed orientation and $\omega$ is a volume form on $S$. Prove that $\omega$ is closed. Prove that $\omega$ ...
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1answer
77 views
+50

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? UPDATE: The surface is embedded in $\mathbb{R}^3$.
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1answer
23 views

Every submanifold of $\mathbb R^n$ is locally a level set

Is it true a very submanifold $M$ of $\mathbb R^n$ is locally a level set? Given a chart $\phi$ about $p \in M$, how can we construct a smooth function $f$ s.t. $f^{-1}(0)= M \cap U$ for some open ...
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-1
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26 views

Curve concatenation in manifolds.

I am having difficulty understanding what is going on geometrically when you add together multiples of curves (1-chains) in a differentiable manifold. Say we have two curves $A$ and $B$ together with ...
7
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1answer
49 views

What the curvature $2$-form really represents?

Let $(E,\pi,B)$ be a principal bundle with structure group $G$. The connection $1$-form can be thought of as a projection on the vertical part. It allows us to characterize the horizontal subspaces as ...
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12 views

Conical surface with negative curvature

I was reading some physics papers and I read about cones possessing negative curvature on the tip (and k = 0 everywhere else). Basically, to build these surfaces instead of removing a sector of the ...
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2answers
32 views

Verification of the identity $\langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle$

In the book Riemannian Geometry, page 91, Do Carmo writes: $$\langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle$$ I could not understand how this happens. Can someone ...
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1answer
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Intuitive meaning of immersions

I have a hard time understanding the concept of immersions. In my course, it was only introduced by the immersion theorem wich says: Let $f: U \subset \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ be ...
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Determinant of the Antipodal Map

Let $f:S^n\to S^n$ be the map defined as $f(x)=-x$. (This map is called the antipodal map). Find the determinant of $df_p$ at a point $p\in S^n$. The sign of determinant should not depend on the ...
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Proving rigorously a map preserves orientation

In the following link: http://hilbertthm90.wordpress.com/2009/09/09/what-i-talk-about-when-i-talk-about-orientation/ They state that the antipodal map $f: \mathbb{S}^{n} \rightarrow \mathbb{S}^{n}$ ...
2
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2answers
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Solving (Frenet-Serret) differential equation system in Matlab

I'm about (trying) to solve the Frenet-Serret equation given by the known formulas, finding $e(s)$, $n(s)$, $b(s)$, where $e'(s) = \kappa(s)v(s)n(s)$ $n'(s) = -\kappa(s)v(s)e(s) + \tau(s)v(s)b(s)$ ...
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1answer
369 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
2
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1answer
24 views

Integrating over an embedded manifold: Jacobian factor?

Let's say I want to integrate a function $$ f(x,y),\quad x\in\Gamma_1,y\in\Gamma_2 $$ where $\Gamma_1,\Gamma_2$ are both embedded manifolds in $\Bbb{R}^3$. The dimension of $\Gamma_1$ is 1 (a ...
5
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1answer
35 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 ...
9
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1answer
309 views

Stokes theorem on Lipschitz-manifolds?

I was wondering if Stokes' theorem could be formulated in a setting which could be easily applied in situations where the traditional form cannot, such as on manifolds with corners like a rectangle or ...
1
vote
1answer
54 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) = ...
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8 views

Cusp-end in the universal covering

Let $M$ be a n-dimensional hyperbolic manifold with finite volume. Then as a consequence of the Margulis-Lemma we have a decomposition in different types of ends. So let $C$ be a cusp-end. Then there ...