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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1answer
25 views

Some relation between parallel vector field and Jacobi field along a geodesic

Cross posted from my question: http://mathoverflow.net/questions/204097/parallel-transport-along-a-geodesic-and-the-related-jacobi-field This is a formula/theorem (written below) that I found ...
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0answers
14 views

Homogeneous Dilation of the Domain in the Free Membrane Problem

Consider the Neumann boundary value problem of the Laplace operator: $$ \begin{cases} \Delta u+\mu u=0,&\text{in }D,\\ \frac{\partial u}{\partial n}=0,&\text{on }\partial D. \end{cases} $$ Let ...
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31 views

Why are algebraic cycles rational?

Let $X_{/\mathbb{C}}$ be a projective non-singular variety of dimension $n$ and $Z \subset X$ be an irreductible subvariety of dimension $p$. Denote by $\mathrm{H}_{\mathrm{dR}}^i(X,\mathbb{C})$ the ...
4
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0answers
20 views

Immersion except at the origin. [on hold]

Whitney showed that for maps of two-manifolds into $\mathbb{R}^3$, a typical cross cap looks like the map $(x, y) \to (x, xy, y^2)$. Prove that this is an immersion except at the origin.
2
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0answers
16 views

What is a local invariant?

Let $(M,g)$ be a Riemannian manifold. Then, it is usually said that $M$ has local invariants associated to $g$. For example, the curvature of the Levi-Civita connection associated to $g$. My question ...
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Find $\nabla_{\gamma'(t)}\gamma'(t)$. Let $S : z = x^2+ y^2$ be a surface in $\mathbb{R}^3$ with the induced metric and let $\gamma(t)$ be

Let $S : z = x^2+ y^2$ be a surface in $\mathbb{R}^3$ with the induced metric and let $\gamma(t)$ be a curve on $S$ given by $\gamma(t) = (t, t, 2t^2)$. For the arc length s$(t) = ...
2
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1answer
20 views

Quick question about covariant derivative

Let $f$ be a function and define $\nabla_X f = X(f)\,\,(1)$, where $\nabla$ is the connection on a manifold and as far as I understand the r.h.s is a function and $X$ is a vector field. I am just a ...
3
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1answer
29 views

Find $\nabla_{\gamma'(t)}\gamma'(t)$. A metric on $\mathbb{R}^2$ is given by the form $dr^2+ f(r,\theta)d\theta ^{2}$ in polar coordinates.

A metric on $\mathbb{R}^2$ is given by the form $dr^2+ f(r,\theta)d\theta ^{2}$ in polar coordinates. Let $\gamma(t)$ be a curve in $\mathbb{R}^2$ given by $\gamma(t) = (t,\theta_0)$ in polar ...
4
votes
1answer
54 views

Is this statement about manifold true? [duplicate]

Suppose $M$ is a closed $k-$manifold in $\mathbb R^n$ without boundary, can we always find a smooth function $f:\mathbb R^n\to\mathbb R^{n-k}$ such that $M$ is the level set where $f=0$?
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0answers
32 views

Forming a (1,1)-tensor field from a (0,2)- and (2,0)-tensor field

Let $A$ and $B$ be 2-tensor fields on a manifold, contravariant and covariant respectively. Prove that there exists a smooth (1,1)-tensor field $C$ with components defined by $$C^i_j = ...
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0answers
15 views

How to parametrize circles on a sphere by the distortion of the equator?

I guess am having a very silly problem right now. Considering a unit sphere $S^2$ and, for example, a curve, in spherical coordinates, $c(t)=(1, \frac{\pi}{2},t)$ that goes around the equator how can ...
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0answers
13 views

inverse function theorem on manifolds

suppose there are two 3-manifolds(consider them as orthogonal matrices $SL(2,\mathbb R)$), and there is $F:SL(2,\mathbb R)\to SL(2,\mathbb R)$, given by $F(A)=A^3$. Can we apply inverse function ...
4
votes
3answers
77 views

Mistake in (Baby) Do Carmo? Elementary topology of surfaces.

If you have the book, it's proposition 2 of section 5.3. If not, the proposition reads: Given any two points p and q $\in$ a regular, connected surface S, there exists a parameterized piecewise ...
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0answers
12 views

What is an infinitesimal automorphism of a connection?

Let $P\to M$ be a principal bundle over a differentiable manifold $M$, with fibre $G$, equipped with a connection $H\subset TP$. I have heard the term "infinitesimal automorphism of $H$" but I haven't ...
3
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2answers
132 views

Is there a well-defined notion of measure zero on topological manifolds?

We extend the concept of measure zero on manifolds by local parameterization. but in this definition we have to check if it is true for every parametrization. In Guillemin's Differential Topology this ...
2
votes
1answer
40 views

A question on integration of differential forms on a manifold

I'm fairly new to differential geometry and have been reading up on integration on manifolds. All the texts/lecture notes that I've read so far always consider integrating an $n$-form over an ...
2
votes
0answers
26 views

Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles.

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
0
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0answers
19 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 ...
1
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1answer
26 views

what is the manifold associated with general linear group? [on hold]

It has dimension n^2 but I want to know the exact manifold structure of general linear group.
4
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0answers
38 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 ...
0
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0answers
14 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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0answers
20 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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0answers
21 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} : = \{ ...
7
votes
2answers
90 views

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 ...
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0answers
22 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 ...
2
votes
1answer
44 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 ...
2
votes
1answer
53 views

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}. ...
1
vote
1answer
20 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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0answers
25 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: ...
-1
votes
0answers
15 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 ...
1
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0answers
53 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 ...
5
votes
2answers
73 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 ...
2
votes
1answer
49 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 ...
0
votes
1answer
34 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 $x=p$, ...
1
vote
1answer
41 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 ...
1
vote
1answer
48 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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
2
votes
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 ...
0
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0answers
24 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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0answers
37 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 ...
0
votes
1answer
29 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 ...
0
votes
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 ...
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votes
0answers
22 views

I know the formula for 4 dimensional curl, what is the formula for 6 dimensional curl? [on hold]

What is the formula for computing the curl of a 6 dimensional vector field?
1
vote
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 ...
1
vote
1answer
25 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 ...
-1
votes
0answers
28 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 ...
1
vote
1answer
40 views

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 $\omega$ ...
0
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0answers
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 ...
-1
votes
0answers
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: ...