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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Linear algebra revisited: What do we do when we set a coordinate system?

I was learning about covariant and contravariant vectors due to special relativity, and it occured to me that we don't live in $\mathbb{R}^4$. I'll explain myself better. Consider the space of ...
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Which of the following are proper patches. (Showing that an inverse of a mapping is continuous)

In which of the following cases is the mapping $\mathbf{x}:\mathbb{R^2} \to \mathbb{R^3}$ a proper patch? (a)$\mathbf{x}(u,v)=(u,uv,v)$ (b)$\mathbf{x}(u,v)=(u^2,u^3,v)$ ...
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1answer
22 views

The relationship between Ricci and Gaussian curvatures

Why do we have that for a surface (dimension $2$) that $$\text{Ric}(X, Y) = K \langle X, Y \rangle ,$$ where $K$ is the Gaussian curvature and $X, Y$ are vector fields?
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22 views

Basic Question on Mayer-Vietoris Sequence

On Pg 449 of Lee's Introduction to SMooth Manifolds (2nd Edition), the Mayer-Vietoris Theorem is given: Let $M$ be a smooth manifold. Let $U$ and $V$ be open in $M$ such that $U\cup V=M$. Then ...
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1answer
36 views

If $ds$ is not a differential form, can I make sense of its intuitive notation somehow?

I understand that a line element is not actually a differential form but a $1$-density. My question is: is the notation $ds^2 = dx^2 + dy^2$ formal in any way? Can it be interpreted as outer or tensor ...
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1answer
26 views

When is the pullback of a tangent bundle along a curve a tangent bundle on the curve?

Consider a smooth manifold $M$; and it's tangent bundle $TM \rightarrow M$; suppose we have curve $c:I \rightarrow M$ When is the pullback $c^*TM$ diffeomorphic to the tangent bundle $Tc$ on $c$? ...
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1answer
14 views

Adjoint representation and tangent vectors

Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra, $\text{Ad}:G\rightarrow GL(\mathfrak{g})$ the adjoint representation of $G$. Then, for $X,Y\in \mathfrak{g}$, \begin{align*} ...
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1answer
49 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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19 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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54 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 ...
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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.
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29 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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19 views

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) = ...
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1answer
25 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 ...
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1answer
32 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. [on hold]

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 ...
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1answer
56 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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34 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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1answer
20 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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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 ...
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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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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 ...
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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 ...
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41 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 ...
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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 ...
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20 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 ...
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1answer
27 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.
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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 ...
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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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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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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} : = \{ ...
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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 ...
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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 ...
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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 ...
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1answer
55 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}. ...
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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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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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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 ...
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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 ...
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2answers
75 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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1answer
51 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
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 $x=p$, ...
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1answer
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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
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 ...
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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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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 ...
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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 ...
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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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38 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
30 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 ...