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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84 views

Is $S^1 \times S^1$ really a torus?

Consider a function $f(x)$ that is $2\pi$ periodic. Consider another function $g(y)$ that is also $2\pi$ periodic. If I wanted to compute the integral of either of these functions I would do so ...
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1answer
32 views

Retraction to the Boundary on Compact Manifold

I was given the following question on an exam today, "Suppose that $M$ is a compact $n$- dimensional oriented manifold with corners. A retraction to the boundary is a continuously differentiable map ...
-3
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1answer
78 views

Can a torus have a simple map from two dimensions to three dimenions like a Gauss map? [closed]

There are several ways to project two dimensions onto a Riemann Sphere and the Gauss map works very well. A Gauss map: 2d {x,y}-> 3d {2*x/(1 + x^2 + y^2), 2*y/(1 + x^2 + y^2), (1 - x^2 - y^2)/(1 + ...
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votes
2answers
26 views

Properties of the ring of smooth function germs, question on proof.

Let us denote by $C_n$ the ring of $C^{\infty}$ smooth function germs $f : (\mathbb R^n, 0) \to \mathbb R$ or the ring of analytic functions germs $f : (\mathbb C^n, 0) \to \mathbb C$. Denote by ...
3
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1answer
93 views

What is the smallest Euclidean space in which one can embed a given curved space?

Given a $d$-dimensional curved space, how many dimensions are required to embed it? As an example think of a sphere's surface, which is a two-dimensional curved space that can be expressed in ...
2
votes
1answer
45 views

Poincaré lemma on a space with trivial homology group

Today I read about Poincaré's lemma from do Carmo's book Differential Forms and Applications. It says that A closed differential $k$-form on a contractible space is exact. I wonder if the ...
5
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1answer
29 views

Are the torsion elements dense in every compact Lie group?

Let $ G $ be a compact connected real Lie group. Denote by $ T $ its set of torsion elements. Is $ T $ always dense in $ G $?
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11 views

Line Elements for $n$-dimensional hyperspheres

I'm currently in the process of deriving the components of the Riemann Curvature Tensor for a 3-sphere using the Cartan Equations. The line element I'm starting with is: $$ ds^2 = ...
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31 views

The inverse image of any regular value is a submanifold

Let $f:M^m\subset\mathbb R^k\to N^n$ be a smooth map between manifolds with $m>n$. Let $y\in N$ be a regular value for $f$. Let $x\in f^{-1}(y)$ and $L:\mathbb R^k\to\mathbb R^{m-n}$ be a linear ...
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0answers
17 views

Definition of frontal map

In the lecture about singularities of curves and surfaces the lecturer gave the following defintion: A smooth map $f: U \subseteq \mathbb R^n \to \mathbb R^m$ is called frontal if and only if for ...
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0answers
34 views

Reference to study moduli spaces

I would like to know about references where the problem of finding the infinitesimal deformations of a given geometric structure, and obtaining the corresponding (elliptic?) complex parametrizing the ...
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1answer
56 views

Is $C = \{(x,y) \mid x^3 + xy + y^3 = c \} \subset \mathbb{R}^2$ an embedded submanifold of $\mathbb{R}^2$?

The problem As a continuation of this question (where it was shown that $C$ was a closed $1$-dimensional submanifold for $c \neq 1/27$), I'm trying to find out whether or not $$C = \{(x,y) \mid x^3 + ...
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3answers
20 views

Greatest area using a string with the length of $l$

Suppose we have a string with length of $l$ what is the shape that has highest area? In other words,with a constant perimeter of $l$ what is the shape with the highest area? P.S:My own speculation ...
5
votes
1answer
73 views

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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0answers
24 views

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)$\mathbf{x}(u,v)=(u^2,u^3,v)$ ...
2
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2answers
43 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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40 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 ...
5
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1answer
65 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 ...
0
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1answer
30 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 $c$? ...
2
votes
1answer
20 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*} ...
3
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1answer
74 views

Relation between parallel vector field along a geodesic and Jacobi field along that same 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
21 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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63 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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27 views

Immersion except at the origin. [closed]

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.
4
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1answer
36 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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20 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) = ...
2
votes
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 ...
3
votes
1answer
42 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. [closed]

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
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1answer
60 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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49 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 = ...
1
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1answer
27 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
15 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
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3answers
79 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
16 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
42 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 ...
3
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0answers
29 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 ...
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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
29 views

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

It has dimension n^2 but I want to know the exact manifold structure of general linear group.
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50 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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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
23 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
22 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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2answers
95 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
26 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
45 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
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1answer
62 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
26 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
28 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
63 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 ...