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
79 views
When is a topological space a manifold?
I'm looking for someone to point me in the direction of papers or books which discuss when a topological space (perhaps with the conditions locally compact, Hausdorff) is a topological manifold, ...
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0answers
42 views
Deformation retract
How to prove that $r_t$ is a deformation retract
$M^a=\lbrace q\in M ; f(q)\leq a\rbrace$
We have the definition :
$r_t$ is a difformation retract if:
$r_t$ is a continius ,onto application ...
1
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1answer
49 views
Show that this is a diffeomorphism
I have a function $F:(0,2\pi) \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}^2$
with $(\phi,r)\mapsto(r(\phi-\sin(\phi)),r(1-cos(\phi)))$ and want to show that this is a smooth(meaning ...
5
votes
1answer
127 views
Constant Rank theorem for domain with nonempty boundary
Problem 4-3 in J.M. Lee's introductory text about smooth manifolds, asks to formulate and prove a version of the constant rank theorem for a map of constant rank whose domain is a smooth manifold with ...
1
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0answers
29 views
How to estimate error pattern of a set of line segments with respect to given reference segments (2D case)
I am having set of pair of line segments (2D). Though each pair should
be coincided on top of each other they are not so. I have been the reference data and then I extracted other line segments ...
0
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1answer
33 views
Nondegenerate critical point
I don't understand this part from the book of Zeidler , can someone help me to understand it ?
Please
Thank you
2
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0answers
28 views
Curvature form projective spaces
Let $T\mathbb{C}P^n$ tangent bundle over $\mathbb{C}P^n$. We have an hermitian metric on $T
\mathbb{C}P^n$ defined as $h=\frac{dzd\bar{z}}{(1+|z|^2)^2}$. If we consider Levi-Civita connection we can ...
1
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0answers
23 views
Relation between triangle and its circumscribed circle, on the surface of a sphere. Generalizations to higher dimensions.
Consider the unit sphere in $R^3$ and an equilateral triangle of side length 1, with all vertices on the surface of the sphere. Now project (from the center of the sphere outward) the triangle and its ...
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0answers
21 views
How can I align the angle between points with the magnetic heading as the points move?
I have 3 robots which must track a point. The distance between all the robots and the point is known so a triangle can be formed between any 2 robots and the point.
If I find the angles in the ...
0
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1answer
33 views
prove that $supp(π^{∗} f) = (supp f)×N.$ Please can you check my answer? Also more explanation please.
My question is that
Let $f \colon M \to R$ be a $C^{\infty}$ function on a manifold $M$.
If $N$ is another manifold and $π \colon M \times N \to M$ is the projection onto the first factor, prove ...
0
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0answers
13 views
The most general form of the metric for a homogeneous, isotropic and static space-time
What is the most general form of the metric for a homogeneous, isotropic and static space-time?
For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) ...
4
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1answer
64 views
How are the isometries $h:(\mathbb{R}^n,||\cdot||_p)\longrightarrow(\mathbb{R}^n,||\cdot||_p)\;$?
An isometry of $\mathbb{R}^n$ is a function $h:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ that preserves the distance between vectors:
$$||h(x)-h(y)||_p=||x-y||_p\;\;, \;\;p\ge1$$
for all $x$ and $y$ ...
1
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1answer
36 views
Why is $\theta \not \in C^{\infty}(S^1)$?
Why is $\theta \not \in C^{\infty}(S^1)$? I know that since $\int_{S^1} d\theta = 2\pi$ then $d\theta$ is not exact. Thus since $d(\theta)=d\theta$, $\theta$ must not be $C^{\infty}$ but it seems ...
2
votes
1answer
64 views
How to show that this set isn't a regular surface?
I'm trying to solve this exercise from Do Carmo's Differential Geometry of Curves and Surfaces, and I want a hint on how to do it. The exercise is:
Is the set $S =\left\{(x,y,z)\in \mathbb{R}^3 \mid ...
1
vote
1answer
105 views
Non unique solution for Ricci flow equation
Why completeness is important for the uniqueness of solution to Ricci flow? For example, if $M$ is the open unit disk in $\mathbb{R}^2$ and $g(0)$ is the Euclidean metric, and hence not complete. Why ...
0
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0answers
96 views
Maple example about Ricci flow collapses the sphere manifold or Einstein manifold to a point in finite time
how to use maple code to demonstrate Ricci flow collapses the sphere manifold or Einstein manifold with positive curvature to a point in finite time
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0answers
62 views
$C_0$ Convergence of Metrics under Ricci Flow when $\chi(M) = 0$.
I am a beginning graduate student, and I am trying to learn basic aspects of Ricci Flow via the uniformization theorem for compact surfaces. I am reading Chow and Knopf's introductory book as well as ...
9
votes
2answers
230 views
Reversing the Ricci flow
Suppose $S$ is a closed, oriented surface (2-manifold) embedded in $\mathbb{R}^3$, which
inherits the metric from $\mathbb{R}^3$, so that distances are measured by shortest
paths on the surface. If ...
2
votes
0answers
113 views
How to prove the holonomy group is preserved under Ricci flow?
I've heard that on a Kaehler manifold $(M,g_0)$, if you evolve the metric $g$ by Ricci flow $\partial g_{ij}(t)/\partial t=-2R_{ij}$, and $g(0)=g_0$, then you always have $g(t)$ is a Kaehler metric on ...
1
vote
1answer
148 views
evolution of curvature under ricci flow , What does the tensor A*B means?
in the book of bennet chow,the definition of the "*" notation is simple ,I can't understand . can someone gives me some examples of this definition? or tell me in which book I can find this?thanks.
0
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0answers
45 views
Strongly parabolic PDE vs weakly parabolic PDE
In my studies on the Ricci flow, I was faced with a problem.
To prove the existence and uniqueness of solutions to the Ricci flow, it is proved that the Ricci flow is a Parabolic PDE type. Then one ...
0
votes
1answer
63 views
Visualize soliton solutions of a PDE
In trying to visualize soliton solutions of a PDE I faced this sentence:
We now think of solitons as self-similar solutions, i.e., solutions which evolve along
symmetries of the flow.
Question 1: ...
2
votes
0answers
49 views
Geodesics and Christoffel symbols
If these are satisfied then we are on a geodesic. Do I just need to plug in the condition given about the christoffel symbols and then see that the equations are allways fulfiled as long as $v=at+b$ ...
2
votes
1answer
56 views
Incomplete vector field
Is there a way I can tell if a vector field on a manifold or just $\mathbb{R}^n$ is incomplete simply by just looking at its formula. For instance on $\mathbb{R}$, $\displaystyle X= (x^2+1) ...
1
vote
0answers
23 views
Equivalence class involving Lie Brackets..
Can anyone help me with a proof, I spent hours on it and nothing: I want to show $$\overline{[fX, Y]}=f\overline{[X, Y]},$$ where $X$ and $Y$ are smooth vector fields on a smooth manifold $M$, $f\in ...
2
votes
2answers
52 views
Extending a smooth map
When can I extend a smooth map $f:\mathbb{R^2}-\lbrace 0 \rbrace \to S^1$ to a smooth map $\tilde{f}:\mathbb{R^2} \to S^1$. For instance, consider $g(x,y)=(x,y)/\sqrt{x^2+y^2}$? Am I able to extend ...
0
votes
0answers
22 views
Laplacian on Reductive coset spaces
Let us consider the sphere $S^n$ (embedded in $\mathbb{R}^{n+1}$) and let $X_i$ denote the vector fields which generate rotations about the $x_i$-axis. My questions are:
(a) Is it true that ...
4
votes
1answer
47 views
How to make a $C^1$ knot into a $C^\infty$ knot
Suppose I have a $C^1$ imbedding $f: S^1 \rightarrow S^3$. From the point of view of knot theory, what's the "best" way to get a $C^\infty$ curve that "looks like" or is "equivalent to" $f$? For ...
2
votes
1answer
42 views
Many partitions of unity on sufficiently “nice”; what does this mean?
In a class I am taking, we are told that are manifolds have "many" partitions of unity (we assume paracompact, Hausdorff, second-countable). However, the course content is not related to this subject. ...
2
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0answers
46 views
Chern classes tangent bundle.
I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over $\mathbb{C}P^n$ ($T\mathbb{C}P^n$) and develop the theory beginning from this ...
1
vote
1answer
25 views
“WLOG” when studying Schwarzschild geodesics
Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\theta=\frac{\pi}{2}$$? I assume it is so because when digging around the internet, most references seem to consider this ...
8
votes
0answers
165 views
Infinite dimensional constant rank theorem
Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
1
vote
1answer
76 views
Möbius maps and their fixed points
Why is it true that if a Möbius map, $f(z)$ fixes distinct $z_1,z_2\in \mathbb C_\infty$ that $f(z)$ either describes a rotation or has a pair of stable and unstable fixed points, such that iterations ...
2
votes
1answer
39 views
Is Whitney sum of vector bundle a categorical colimit?
We known that the direct sum of two vector spaces is the categorical colimit of vector spaces. My question is whether Whitney sum of vector bundle is a categorical colimit (in the category of vector ...
5
votes
2answers
98 views
How to show something is a contraction?
If we let $X$ be a complete metric space, and let $S:X\to X$ be a map, such that $S^m$ is a contraction. We now want to show, that $S$ has a unique fixed point
This is what I've thought so far:
Due ...
5
votes
1answer
56 views
Triangulation of a manifold adapted to a submanifold
I am not extremely proficient in topology, and am concerned with the following question :
Given a compact manifold $X$ and a submanifold $Y \subset X$, is it always possible to find a triangulation ...
6
votes
1answer
259 views
Cigar soliton solution
In wikipedia, it has been written that an important 2-dimensional example of Ricci flow over $M=\mathbb{R}^2$ is given by $g((x,y),t)=\frac{dx^2+dy^2}{e^{4t}+x^2+y^2} \;\;\; (\star) $
Here are my ...
2
votes
1answer
42 views
Help with algebraic manipulation to prove that $\Omega (M)$ constructed from $\Omega^1 (M)$ forms an algebra over $C^\infty (M)$
In Baez´s Gauge Theories, Knots and Gravity he states that the differential forms on a n dimensional manifold M, $\Omega (M) = {\bigoplus}_p \Omega^p (M)$, constructed from $\Omega^1(M)$ and the ...
3
votes
1answer
67 views
Basic question about definition of Chern classes
Apologies if there is something I missed with a quick internet search, but why do we define Chern classes for complex vector bundles (instead of real vector bundles for example)?
If we define chern ...
1
vote
1answer
25 views
Curvature (Gaussian) of a hypersphere
I am looking for a general formula for the Gaussian curvature of an $n$-sphere (the set of points in $R^{n+1}$ equidistant from the origin) of radius $r$.
From what I have read, there would be $n$ ...
2
votes
1answer
67 views
“Completing” a vector field on a non-compact manifold $M$
Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete.
Is there a way to create a smooth vector field $V$ that is ...
0
votes
1answer
38 views
the osculating planes of a curve pass through a fixed point $\rightarrow$ the curve is a plane curve.
If the osculating planes of a curve pass through a fixed point, the curve is a plane curve.
How to prove it?
4
votes
1answer
58 views
Differential of smooth function on manifold
In the book I am using, the author defines differentials in the following way.
Given smooth manifolds $M,N$ and a smooth mapping $\psi:M\to N$ define the differential $d\psi_m$ at a point $m\in M$ as ...
0
votes
2answers
37 views
What is the initial reason to define the evolute of a curve?
The evolute of a curve is defined as the envelope of the normals or as the locus of the center of the osculating circle.
What is exactly "the envelope of the normals" ?
What is the reason to ...
1
vote
1answer
89 views
Complete non-vanishing vector field
Let $M$ be a non-compact smooth manifold. Suppose we have a nowhere-vanishing smooth vector field X. Is this vector field complete?
I know it is when $M$ is compact. However, I am unsure in the ...
0
votes
0answers
19 views
Cotangent bundle of a complex projective space
How does the cotangent bundle of a complex projective space looks like? Is that an Einstein manifold?
0
votes
1answer
25 views
Surfaces of Constant Gaussian Curvature
I'm preparing for an exam and I would like to know what are some examples of surfaces with constant Gaussian curvature such as surfaces with $k=0, \pm1$
2
votes
0answers
33 views
Submanifold with boundary of a manifold with boundary
Let $M$ be a smooth manifold.
(1) A subset $S$ of $M$ that with the subspace topology is a topological manifold (with or without boundary), together with a differential structure that makes the ...
1
vote
1answer
36 views
Implicit Function Theorem and Rank Theorem Misunderstandings.
Regular values are useful because of the generalization of the first part of the implicit function theorem: if $q$ is a regular value of $f:M \to N$ (with dimension $m$ and $n$ respectively), then $$A ...
3
votes
0answers
66 views
Second derivative of a metric in terms of the Riemann curvature tensor.
I can't see how to get the following result. Help would be appreciated!
This question has to do with the Riemann curvature tensor in inertial coordinates.
Such that, if I'm not wrong, (in inertial ...
