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 ...
32
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8answers
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Teaching myself differential topology and differential geometry
I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.
I have decided to fix this lacuna once for ...
2
votes
2answers
339 views
About connected Lie Groups
How can I prove that a connected Lie Group is generated by any neighborhood of the identity?
The result is almost trivial for $R^n$ but I tried using the open subgroup generated by this ...
48
votes
4answers
4k views
Why is a circle in a plane surrounded by 6 other circles
When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other number?
I'm ...
23
votes
3answers
1k views
Why are smooth manifolds defined to be paracompact?
The way I understand things, roughly speaking, the importance of smooth manifolds is that they form the category of topological spaces on which we can do calculus. The definition of smooth manifolds ...
7
votes
1answer
207 views
Finding a smooth function less than some given (positive) continuous function
Let $M$ be a smooth manifold ($dim\ge 1$). Let $f:M\to\mathbb{R}$ be a positive continuous function. Prove there is a smooth map $g\in C^{\infty}(M)$ such that $0<g<f$.
I knew this would ...
21
votes
2answers
1k views
Which manifolds are parallelizable?
Recall that a manifold $M$ of dimension $n$ is parallelizable if there are $n$ vector fields that form a basis of the tangent space $T_x M$ at every point $x \in M$. This is equivalent to the tangent ...
28
votes
3answers
2k views
Why is the Möbius strip not orientable?
I am trying to understand the notion of an orientable manifold.
Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
22
votes
3answers
567 views
Is every Compact $n$-Manifold a Compactification of $\mathbb{R}^n$?
I read the result that every compact $n$-manifold is
a compactification of $\mathbb{R}^n$.
Now, for surfaces, this seems clear: we take
an n-gon, whose interior (i.e., everything in
the n-gon except ...
11
votes
5answers
673 views
Why the interest in locally Euclidean spaces?
A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds).
What is the special feature of Euclidean spaces that makes them interesting?
...
5
votes
2answers
382 views
How to Visualize points on a high dimensional (>3) Manifold?
Are there any ways to visualize(plot/draw) points on a high dimensional (ex: dimension = 5) manifold?
4
votes
2answers
176 views
Are there any compact embedded 2-dimensional surfaces in $\mathbb R^3$ that are also flat?
Let $\overline{g}$ be the flat metric on $\mathbb{R}^3$.
I would like to know if there is any compact embedded 2-dimensional surface $M$ in $\mathbb{R}^3$ (without boundary) such that ...
3
votes
3answers
133 views
Topological boundary vs geometric boundary
Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$
$M_2=\{(x,y) \mid x^2+y^2\le1\}$
What are the interior of $M_1$ and $M_2$ ?
And What are the boundary of $M_1$ and $M_2$ ?
How to find them? ...
0
votes
2answers
81 views
Lie bracket in local coordinates. Find the formula $c^{k}$ in terms of $a^{i}$ and $b^{j}$
This is from T.U Loring's manifold book. I tried. But I didnt do the question. Please show me how to solve instructively and explicitly. I want to learn this topic. Thank you for help.
0
votes
1answer
55 views
The differential $i∗ : TpS_{2} → TpR_{3 }$ maps $ ∂/∂u|p,∂/∂v|p $ into $TpR_{3}. $ Find $(α_{i}, β_{i}, γ_{i})$
Hi! This was my homework. Prof. sent its answer. But I didnt understand how can this answer be reached? Please can someone explain this?
3
votes
2answers
290 views
An application of partitions of unity: integrating over open sets.
In Spivak's "Calculus on Manifolds", Spivak first defines integration over rectangles, then bounded Jordan-measurable sets (for functions whose discontinuities form a Lebesgue null set).
He then uses ...
18
votes
10answers
2k views
“Immediate” Applications of Differential Geometry
My professor asked us to find and make a list of things/facts from real life which have a differential geometry interpretation or justification. One example is this older question of mine. Another ...
16
votes
2answers
319 views
Roadmap to study Atiyah-Singer index theorem
I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
16
votes
2answers
457 views
Relationship between the zeros of a vector field and the fixed points of its flow
I'm having a little trouble here and would appreciate some hints.
Let $M$ be a compact manifold without boundary and let $X$ be a smooth vector field on $M$ with only isolated zeros. Let $\theta_t$ ...
4
votes
3answers
469 views
meaning of dual space of a tangent space?
We know that a tangent vector is a directional derivative operartor, and the collection of all tangent vectors at a point is a tangent space. I don't understand the intuitive meaning behind the dual ...
6
votes
1answer
143 views
How to Classify $2$-Plane Bundles over $S^2$?
I'm curious how one can classify the bundles over a given manifold. I recently read this paper on classifying $2$-sphere bundles over compact surfaces. A lot of the concepts went over my head since ...
4
votes
5answers
605 views
Differential Geometry of curves and surfaces: bibliography?
Dear all, next year, I will probably teach a one-semester course of Differential Geomtry of curves and surfaces. Its content must be something along the lines of the first four chapters of Do Carmo's ...
14
votes
1answer
385 views
Group cohomology versus deRham cohomology with twisted coefficients
Let $G$ be a simple simply-connected Lie group, let $M$ be a 3-manifold and $P \to M$ a principal $G$-bundle. Let $A$ be a flat connection in this bundle, and let $\text{Ad} P$ be the associated ...
7
votes
2answers
492 views
Are there higher-dimensional analogues of sectional curvature?
I recently learned that on Riemannian manifolds, one can define the sectional curvature (http://en.wikipedia.org/wiki/Sectional_curvature) of a (2-dimensional) plane section. I was wondering if a ...
6
votes
1answer
227 views
embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$
Consider the classic map
$$F:\mathbb{RP}^2\rightarrow \mathbb{R}^4$$
defined by $$F[x,y,z]=(x^2-y^2,xy,xz,yz)$$.
This defines a smooth embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$. It is clearly ...
4
votes
0answers
169 views
Geodesics on the torus
[This is a follow-up to my question Is there a Möbius torus?]
Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus:
There are five ...
11
votes
2answers
375 views
Is $M=\{(x,|x|): x \in (-1, 1)\}$ not a differentiable manifold?
Let $M=\{(x,|x|): x \in (-1, 1)\}$. Then there is an atlas with only one coordinate chart $(M, (x, |x|) \mapsto x)$ for $M$. We don't need any coordinate transformation maps to worry about ...
5
votes
1answer
154 views
Extension of Riemannian Metric to Higher Forms
I've been reading about Riemannian manifolds, and have come across a comment that says that for a metric $g$ on an $N$-dimensional manifold $M$, considered as a bilinear map
$$
g:\Omega^1(M) \times ...
4
votes
1answer
334 views
Problem book on differential forms wanted
I want to get used to differential forms. Thus I would like to solve a bunch of problems, especially on integration of differential forms.
So I need a collection of problems with answers/solutions, ...
3
votes
1answer
538 views
Why can we think of the second fundamental form as a Hessian matrix?
Let $f: U \rightarrow \mathbb{R}^3$ be an immersion that parametrizes a piece of a surface, and let $(h_{ij})$ be the matrix for the second fundamental form of that surface.
According to pg. 70 of ...
2
votes
0answers
74 views
Show that the projection map is Orientation preserving iff n is even
My question is that
Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere
$U =${$x∈S^n |x^{n+1} >0$}.
It is a coordinate chart on ...
2
votes
2answers
81 views
What is the limit distance to the base function if offset curve is a function too?
I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that ...
2
votes
1answer
410 views
Properly Defining a Smooth Curve
I have seen many different definitions of what it means for a curve to be "smooth". In this question, for instance, a curve $\gamma \colon [a,b] \longrightarrow \mathbb{R^n}$ is defined to be smooth ...
2
votes
1answer
889 views
Classsifying 1- and 2- dimensional Algebras, up to Isomorphism
I am trying to find all 1- or 2- dimensional Lie Algebras "a" up to
isomorphism. This is what I have so far:
If a is 1-dimensional, then every vector (and therefore every tangent
vector field) is ...
5
votes
1answer
142 views
On minimizing the area of an enclosing surface subject to nonnegative Gaussian curvature
This is inspired by this previous question on physical processes that might give rise to convex hulls.
Consider the problem of gift-wrapping a three-dimensional object using an inextensible material, ...
4
votes
1answer
282 views
Topology needed for differential geometry
I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know? I know some basic concepts reading from ...
3
votes
1answer
239 views
Lie Algebra Homomorphism Question
So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
3
votes
2answers
212 views
Total Derivative and Multilinear Functions
So I'm starting to work through Spivak's Calculus on Manifolds and I'm having a little trouble verifying some of the claims made in the book problems. To review:
Given a function ...
3
votes
1answer
264 views
Tangent space to circle
I guess I am missing something obvious here. I am reading about vector bundles. (What Karoubi calls 'Quasi Vector-Bundles')
An example is the sphere, where for every point $X \in S^n$ we choose $E_X$ ...
2
votes
2answers
329 views
Reconstructing space curves from its curvature and torsion
I have to write a program which gets two functions (curvature and torsion) and 3 vectors of the Frenet-Serret "frame" at the starting point - and I have to reconstruct the space curve from this given ...
2
votes
2answers
187 views
Orientation on $\mathbb{CP}^2$
I am confused by the orientation of a topological manifold.
My understanding is: An orientation of a topological manifold is a choice of generator of the $H^n(M,\mathbb Z)$. So given a manifold, we ...
2
votes
0answers
46 views
Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface
This question is sort of an extension to this previous question of mine,
Hyperellipticity (or not!) of a Riemann surface and the singularities of the curve
If one knows the multiplicity of a ...
2
votes
1answer
258 views
planar curve if and only if torsion
Again I have a question, it's about a proof, if the torsion of a curve is zero, we have that
$$ B(s) = v_0,$$
a constant vector (where $B$ is the binormal), the proof ends concluding that the ...
2
votes
2answers
373 views
Direction of the second derivative of an arclength parametrized curve
I have a question, it's so simple and stupid ._. If I have a planar curve parametrized by arc length, it's easy to show that the second derivate is orthogonal to the first derivate vector (tangent ...
1
vote
1answer
142 views
shortest distance between two points on $S^2$
Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$
Length of a curve in $3D$ is ...
1
vote
1answer
388 views
Prove that curve with zero torsion is planar
I have proved that a planar curve of zero curvature is a straight line. It follows from the Frenet equations.
But now I need to prove that if $\varkappa=0$, then the space curve $\mathbf{r}(t)$ is ...
0
votes
1answer
54 views
Can any smooth planar curve which is closed, be a base for a 3 dimensional cone?
A cone in 3 dimensions has a vertex and a base. The contour of the base is a circle which is a smooth closed planar curve. Can there be a more general cone which can have any smooth closed planar ...
24
votes
4answers
1k views
Is there any easy way to understand the definition of Gaussian Curvature?
I am new to differential geometry and I am trying to understand Gaussian curvature. The definitions found at Wikipedia and Wolfram sites are too mathematical. Is there any intuitive way to understand ...
56
votes
2answers
2k views
Direct proof that the wedge product preserves integral cohomology classes?
Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$.
There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients ...
8
votes
1answer
965 views
Intuitive explanation of covariant, contravariant and Lie derivatives
I would be glad if someone could explain in intuitive terms what these different derivatives are, and possibly give some practical, understandable examples of how they would produce different results.
...
13
votes
1answer
1k views
Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?
the Fourier transformation of a scalar function with respect to one variable might be defined as
$\mathcal{F}\left[w\right](\omega )\equiv ...
