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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32 views
Show that ${\alpha}$ is a line of curvature if and only if ${\alpha}$' is parallel to (Gradient of U in direction of (alpha)') along ${\alpha}$
A curve on M is a line of curvature if ${\alpha}$(t) is an eigenvector of the shape operator
for all t.
This is equivalent to saying that the unit tangent vector T(alpha) is a principal
vector.
...
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3answers
102 views
Prove that Gauss map on M is surjective
Let $M$ be a closed, orientable, and bounded surface in $\mathbb{R}^3$.
(a) Prove that the Gauss map on $M$ is surjective.
(b) Let $K_+(p) = \max \{0, K(p)\}$. Show that
$$
\int K_+dA \ge 4\pi.
$$
...
2
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1answer
43 views
Prove Green formula
Let $(M^n,g)$ be an oriented Riemannian manifold with boundary $\partial M$.
The orientation on $Μ$ defines an orientation on $\partial M$. Locally, on the boundary, choose a positively oriented ...
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1answer
35 views
connections on an embedded sumanifolds of Euclidean space
I would like to see how to derive the connection 1 form on the bundle of oriented orthonormal frames for a manifold embedded in Euclidean space.
It seems that the geometry resides in a collection of ...
1
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0answers
24 views
Trivialization of a path of tamed almost complex structures
I am wondering if the following result is true:
Let $(V,\omega)$ be a symplectic vector space and $\{J_t\}_{0\leq t\leq 1}$ a smooth path of almost complex structures on $V$ which are tamed by ...
2
votes
1answer
47 views
Boundary orientation for a cylinder
Please help me.I am think that I can use stokes theorem but ı could not apply.This question is very benefical for me to learn the subject please help me :(
2
votes
1answer
44 views
How to conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$?
Can anyone explain to me how I can conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$ by using integration by parts and $\langle f_1 ,f_2 \rangle_\mu:=\int_M f_1 f_2 d\mu$?
Where $M$ is a ...
3
votes
1answer
39 views
Smoothness in Banach space
I need a reference about a definition. Let $n$ be an integer and $G$ be a group of $H^n$(Sobolev) automorphisms of a vector bundle $E$ on some manifold $M$ and $C$ be the space of connections of class ...
2
votes
0answers
36 views
Orientation-preserving diffeomorphism [duplicate]
Can you help for solving this please. Although I study this subject I could not solve this question please help me ı am willing to learn this question.
1
vote
0answers
13 views
Metric spaces and curvature
Suppose that it is given that the Riemann curvature tensor in a metric space of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the ...
1
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0answers
26 views
(Basic) question regarding Einstein-Hilbert-functional / total scalar curvature
I got a question regarding the total scalar curvature / Einstein-Hilbert-functional.
I found it in Peter Topping's book "Lectures on the Ricci flow" (you can download it for free here: ...
2
votes
2answers
66 views
How does the differential $df$ act on an element of $T_pM$?
Let $f$ be a smooth real valued function on a smooth manifold $M$. The differential of $f$ is the covector field $df$ defined by
$$df_p(v) = v(f)$$
where $v \in T_pM$ and where we are now thinking ...
1
vote
1answer
33 views
Now I am asking that the topological and manifold boudary for real line I am grateful to explain me more clearly and instructively.
Let M be the subset $[0,1[$ $∪ $ {$2$} of the real line. Find its topological boundary $bd(M)$ and its manifold boundary $∂ M$.
I know that while I find the topological boundary, I need to show ...
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34 views
on flows of two vector fields
this is the last part of a homework question. I got some problem understanding the question itself, wondering if anyone can help me with this part.
On manifold $R^2-\{0\}$, define two vector fields ...
1
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0answers
28 views
Proof of Lie theorem on solvable Lie algebra
I am reading a book of Helgason.
As you know, solvable Lie algebra $g \subset V= {\bf C}^n$ have a nonzero $v$ such that
$v$ is an eigenvector of any element of $g$.
I can follow the proof in ...
1
vote
1answer
60 views
Coordinate transform
Can anyone see what transformation $$r\to f(r)$$ transforms $$\exp(2\phi(r))(dr^2+r^2d\theta^2)$$ to
$$df^2+\sinh^2(f)d\theta^2$$?
I there a systematic way to attack such a problem -- rather than just ...
1
vote
0answers
19 views
A better way to see this relation concerning Ricci tensor components
If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
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votes
1answer
35 views
least square solution for obtaining a line 3d by intersecting many planes
If I have been given 4 planes and I know only a point (just a point lie on the plane e.g. o1,o2, o3 & o4) and normal vector of each plane (n1, n2, n3 & n4). Then, by intersecting all 4 (or ...
3
votes
3answers
132 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? ...
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vote
2answers
45 views
Parametrization of $n$-spheres
This comes from Guillemin and Pollack's book Differential Topology. The book claims that one cannot parametrize a unit circle by a single map. I thought we could (by a single angle $\theta$).
I ...
4
votes
1answer
69 views
The open Möbius Band is not orientable
Can you explain my green underlying please.I have confused and ı dont understand why by the continuity of the orientation at the points $(0,0)$ and $(1,0)$ are also $e_{1},e_{2}$
4
votes
2answers
83 views
Orientations on Manifold
This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
2
votes
1answer
51 views
Spherical geometry - relating angles of lunes and segments of great circles
Consider the picture below. I have a sphere of radius $r$, centered at $C$. The angle $\varphi$ is the dihedral angle between the plane defined by the shaded area and a plane through the indicated ...
2
votes
1answer
81 views
Why is the cylinder surface on $\Bbb R^3$ orientable?
Why is the cylinder surface on $\Bbb R^3$ orientable? Please can someone explain me clearly?
1
vote
1answer
32 views
How to find the gradient for a given discrete 3D mesh?
I have a 3D mesh that is looking like this:
ie I have a set of triangles in a 3D space, and they are all linked by their edge.
I have to compute the gradient associated with this field, at each edge ...
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votes
1answer
68 views
Orientation preserving diffeomorphism.
I am stuck with the question. I guess that I need to write jacobian matrix. But I could not do. Please help me thank you
0
votes
1answer
43 views
integral of closed differential form
This is my first course in differential forms so it might be a trivial question. If $\mu$ is a $n-1$-form on $n$-dim manifold $X$ the book uses that
$$\int_X{d\mu}=0.$$
Is this expression valid for ...
3
votes
3answers
68 views
Transverse intersection of multiple submanifolds
Let $M$ be a smooth manifold and suppose that we have three (or more) submanifolds $N_1,N_2,N_3\subset M$.
What is the right notion of "transverse intersection" of $N_1,N_2,N_3$, i.e. what is the ...
4
votes
2answers
57 views
Proving homeomorphism between surface and $\mathbb{R}^2$ minus Cantor Set
I've been working with Spivak's Differential Geometry exercises and I found myself confused with this one: "Let $C\subset \mathbb{R} \subset \mathbb{R}^2$ be the Cantor set. Show that $\mathbb{R}^2 - ...
1
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1answer
58 views
Diffeomorphism of open intervals in $\mathbb{R}$ with specified values
I know two open intervals on $\mathbb{R}$ are diffeomorphic to each other. My question is if I have a intervals $(a-\varepsilon,b+\varepsilon)$ and $(c-\delta,d+\delta)$, is there a diffeomorphism ...
2
votes
0answers
58 views
Exponential Map
this seems to be an easy question but I'm stuck anyway. Let $\Gamma$ be a submanifold of a Riemannian manifold $(M,g)$. Let further $U$ be a coodinate neighborhood
of $\Gamma$ such that a point ...
2
votes
0answers
28 views
Why can I write the curve shortening flow system as..
Consider the family of curves $$\gamma:S^1\times [0, T)\rightarrow \mathbb R^2, \gamma(u, t)=(x(u, t), y(u, t)),$$ where $u$ parametrizes the trace of the curve and $t$ parametrizes which curve is ...
2
votes
1answer
58 views
Why is the diffential of a map between manifolds a map between the tangent spaces?
In the books that I have seen, given a smooth map $\phi: M \rightarrow N$ where $N$ and $M$ are manifolds, the differential at a point $x$ is defined as $d \phi_x: T_x M \rightarrow T_x N$. Why is it ...
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votes
0answers
24 views
Imaginary line passing through non-collinear points in R3.
I have come to a problem where n points are provided in 3-Dimensional plane. I need a imaginary line which can be assumed that it is passing through these points.
1
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0answers
40 views
What is the needed background to study tensors?
What is the needed background to study tensors?
I do not plan to study it but, I want to know the background!
It's a matter of curiosity .
Thanks.
1
vote
1answer
34 views
if the curvature is constant and positive, then it is on the circunference
I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the ...
0
votes
1answer
31 views
What are some geodesics of the metric $ds^2=\frac{1}{y^2}(dx^2+dy^2)$?
Ok, we have the metric $ds^2=\frac{1}{y^2}$ defined in the upper half plane $U=\{(x,y)\in\mathbb{R}^2|y>0\}$.
I know two geodesics are $x(t)=a-b\cos{t}$ and $y(t)=b\sin{t}$. What are some others? ...
1
vote
1answer
22 views
The continuity of principal coordinate system
$X$ is a $C^k$ hypersurface in $\mathbb R^{n+1}$ and $y$ is a fixed point on $X$. Can we find an orthogonal system $\{e_1(x),e_2(x),\cdots,e_{n+1}(x)\}$ on a neighborhood $U$ of $y$ such that 1. ...
1
vote
1answer
61 views
Show that the zero set of $f$ is an orientable submanifold of $\Bbb R^{n+1}$.
Suppose $f(x_1,...,x_{n+1})$ is a$ C^∞$ function on $\Bbb R^{n+1}$ with $0$ as a regular value. Show that the zero set of $f$ is an orientable submanifold of $\Bbb R_{n+1}$. In particular, the unit ...
0
votes
1answer
32 views
How to decide whether F is orientation-preserving or orientation-reversing as a diffeomorphism onto its image.
Let $U$ be the open set $(0,∞)×(0,2π)$ in the $(r,θ)$ -plane $R^2$.
We define $F : U ⊂ R^2 → R^2$ by $F (r, θ ) = (r cos θ , r sin θ )$.
How to decide whether F is orientation-preserving or ...
1
vote
1answer
39 views
Obvious Killing vectors?
What obvious Killing vectors do these metrics have?
(a) $ds^2=\frac{1}{y^2}(dx^2+dy^2)$, $-\infty<x<\infty,y>0$
(b) $ds^2=d\mathscr{X}^2+\sinh^2{\mathscr{X}}d\phi^2$, ...
2
votes
0answers
71 views
Figure $\infty$ is immersion of circle
Where can I find prove of:
Figure $\infty$ is immersion of circle.
More thanks for a prove or a function between these manifolds.
3
votes
0answers
60 views
Kähler Geodesics
Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric
$$\begin{pmatrix} ...
2
votes
1answer
32 views
The connectivity of the intersection of hypersurface and ball
$u$ is a function defined on a connected open set $\Omega$ of $\mathbb R^n$ containing $0$ such that $u \in C^2(\Omega)$ and $u(0)=0$. Consider the hypersurface $X=\{(x,u(x))~|~x\in\Omega\}$ and the ...
3
votes
1answer
48 views
Determining the embedding space:
I have seen a lot of discussion of alternate geometries for example on a sphere or hyperbolic saddle as opposed to a plane:
Has anyone consider the notion of that plane or hyperbolic saddle itself ...
2
votes
1answer
55 views
$F(x,y) = (x^2 +y^2,xy)$. compute $F^{∗}(u \, du+v \, dv)$
Let $F : \Bbb R^2 → \Bbb R^2$ be given by
If $u$,$v$ are the standard coordinates on the target $\Bbb R^2$,
compute $F^{∗}(u \, du+v \, dv)$.
$$F(x,y) = (x^2 +y^2,xy).$$
I am confused so much. I ...
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votes
2answers
40 views
How can I show that if the second fundamental form of a surface is identically equal to zero, then the surface is a plane?
This is my question:
Let P be a plane considered as a surface in 3-space. Show that its second fundamental
form is zero. Conversely, show that if the second fundamental form of a surface is ...
0
votes
1answer
49 views
Partitions of unity and bump function
I can not image this guestion in my mind.can you give me graph and help how ı can prove this question please.
1
vote
1answer
42 views
Willmore energy of an ellipsoid
Given an ellipsoid of equation
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$
How can I calculate the Willmore energy of this surface knowing that its definition is:
...
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votes
1answer
45 views
A question about Moebius strip
The Moebius strip (without boundary) $ S $ can be realized as a regular surface of $ R^3 $ (regular surface is meant in the sense of do Carmo's book). Using a 'manifold' language it can be proved that ...
