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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Compactness property
Let $\Omega \subset X$, X: Banach space. Given $\varepsilon \ge 0$, we define the set of $\varepsilon-normals$ to $\Omega$ at $\bar{x}$$\in \Omega$ by:$\widehat N_\varepsilon(\bar ...
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Differentiable manifolds, Serge Lang
I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...
5
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0answers
62 views
Invariant submanifolds
Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
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Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
2
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2answers
47 views
Smooth maps on a manifold lie group
$$
\operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\
\begin{align}
&n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\
&n = 2, \operatorname{GL}_n(\mathbb ...
2
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Alternative rigorous definition of a surface integral
Consider some open subset $U$ of $\mathbb{R}^n$ where $U$ has a (piecewise) $C^1$-boundary. Let $f$ be some smooth (enough) real function. Is there some way to give a measure-theoretic definition of ...
4
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1answer
29 views
Complexified tangent vector, complex manifold
Consider a complex submanifold $M$ of a complex ambient vector space $X$. Suppose you have a base point $p \in M$ and a $C^1$ arc $\gamma(t)$ passing through $p$ and staying in $M$, with tangent ...
6
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1answer
48 views
Hamiltonian for Geodesic Flow
I'm trying to prove that geodesic flow on the cotangent bundle $T^* M$ is generated by the Hamiltonian vector field $X_H$ where
$$H = \frac{1}{2}g^{ij}p_i p_j$$
but I am stuck. Could somebody show ...
1
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1answer
31 views
Space of embedded surfaces with a common point
Consider the space of all embedded orientable surfaces in $ R^3 $ of constant mean curvature (the minimal case is included) passing through the origin. I'm asking if there exists a topology on this ...
1
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2answers
39 views
What does generic immersion mean?
I have been looking for the meaning of generic immersion
In the textbook I am reading, a theorem involves a curve with y coordinate satisfying $y'(0)=0$ says at some stage the following: "...Since ...
2
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2answers
53 views
Symplectic Form Preserved by Orthogonal Transformation
I'm trying to prove that the symplectic form
$$\omega = d(\cos\theta) \wedge d\phi$$
is preserved by the action of $SO(3)$ on $S^2$ where $\phi$ and $\theta$ are spherical polars. Now $SO(3)$ simply ...
2
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0answers
31 views
No flux boundary condition on PDE on surface (Laplace-Beltrami)
What would a Neumann BC on a PDE posed on a surface look like? In the flat case, we have
$\nabla u \cdot N = 0$
where $u$ is the solution of the PDE and $N$ is unit normal vector.
In a surface case, ...
1
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1answer
38 views
What is a conormal vector to a domain intuitively?
I read that a conormal vector of a domain is a vector that is tangential to the domain and normal to its boundary.
If we consider an open disk in $\mathbb{R}^2$ what is a conormal vector at a point ...
1
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1answer
51 views
Every curve is a geodesic??
I've been reading up on how isometries send geodesics to geodesics. I recently saw a proof of another theorem that used the fact:
The set of fixed points of an isometry is a geodesic.
But isnt the ...
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1answer
11 views
Inequality in proving the Isoperimetric Inequality
The question came from reading the following post: A proof of the Isoperimetric Inequality - how does it work?
I almost can follow the whole proof, but I am stuck at one point. Why does $(x^2 ...
1
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1answer
22 views
How to directly show that Figure 8 injective immersion is not a monomorphism
I'm working in the category of smooth manifolds. The injective immersion that takes the open unit interval $(0,1)$ to the figure 8 is a well-known example of an injective immersion that is not an ...
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2answers
96 views
The euclidean space $\Bbb R^n$ is orientable as a manifold.
I know that
The euclidean space $\Bbb R^n$ is orientable as a manifold.
I think that it is orientable because it has a nowhere vanishing $n$-form.
But I am not sure.
Please can you explain ...
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0answers
22 views
maxima of the sum of unimodal functions .
I have a set of unimodal functions. Each function has real roots. All roots of each function lie outside a certain limit points. These limit points are the same for each function. Each function is in ...
5
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0answers
40 views
What is the volume of Complex Projective Space with Fubini-Study Metric?
I try to compute the volume of the complex projective space $\mathbb{CP}^n$ with Fubini-Study metric, normalized to have diameter $=\pi/2$ i.e. the sectional curvatures lie between $1$ and $4$. Fix a ...
4
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1answer
49 views
Applications of information geometry to the natural sciences
I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
2
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0answers
48 views
Differential Geometry Video Lectures
I know there's a similar question here, however since what I found there wasn't what I was looking for I thought on creating a new question. I'm studying Differential Geometry through Spivak's book "A ...
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0answers
27 views
Following a polyline along the surface of a polygon that is twisted
I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource.
For my problem I need to get the ...
2
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0answers
33 views
Stoke's theorem application to curl theorem. I did. Please can you check it?
Now, I need to apply stoke's theorem to curl theorem.
My teacher gave a hint.
Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$
$dim(M)=2$
M is the subset of $\Bbb ...
2
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0answers
50 views
Real projective space is Hausdorff
I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix??
This prove is correct or I need to add something ?? ...
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1answer
25 views
how to calculate $d\Omega(f)$ here
the question was to find $d \Omega(f)$ with :
$$ \Omega : (E,[.]) \to (F,||.||) \\f \to -f'' +f^3$$
$
[f] = |f'(0)| + ||f''|| $
; $ ||f|| = Sup_{[0,1]}|f(x)| $
the answer is given to me like this ...
3
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4answers
73 views
Curvature of a non-compact complete surface
Assume $\Sigma$ is a non compact, complete surface. Assume the integral $$\int_{\Sigma}K$$ is convergent, where K is the Gauss curvature of $\Sigma$. Is it always true that ...
2
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1answer
57 views
Arc length parameter s
Consider the metric $$ds^2 = \frac{dx^2+dy^2}{y^2}.$$
Assume $R>0, a\in\mathbb{R}$. Consider the curve $$\gamma(\theta)=(a+R\sin\theta,R\cos\theta)$$
for $-\frac{\pi}{2}\leq\theta\leq ...
1
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1answer
53 views
Real Projective Space
How I can prove this corollary 7.15.I consider that ı can apply the corollary 7.10.But I could not can you help me.
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2answers
88 views
Curvature of a curve whose unit tangent creates a constant angle with z-axis
I have this question:
Consider a curve $\gamma$(s) and its projection to the plane $\beta$(s), i.e
$$\gamma(t) = \begin{bmatrix} f(s)\\ g(s) \\ h(s) \end{bmatrix}, \beta(t) = \begin{bmatrix} f(s) \\ ...
1
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1answer
25 views
Lie subalgebra, Lie subgroup and membership
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $H$ be a connected Lie subgroup with Lie algebra $\mathfrak{h}$.
We have that $X \in \mathfrak{h} $ iff $exp(tX) \in H \ \ \ \forall t ...
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1answer
31 views
Locally finite or not
I am tryıng to learn locally finite and can you give an explanation for my green writing please thank you
3
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1answer
72 views
What is overlop
I want to ask something that what is overlop.My teacher said that For Ex1, Everything is overlop hence it is not locally finite.For example 2,it doesnt overlop.Actually ı could not understand.please ...
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0answers
47 views
Topological manifold example
$\theta(x,x^2)=x$
$\Bbb X =${$(x,x^2)| x$ in $\Bbb R$}
And V is subset of $\Bbb R$
$dim\Bbb X=1$
My instructor said that this is topological manifold.
Why?
Please can you explain me? This ...
0
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0answers
41 views
Geodesic equation for a 2D manifold
I am having trouble understanding how the following statement (taken from some old notes) is true:
For a 2D manifold such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$
If we assume that $$\dot x^a\dot ...
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2answers
54 views
An open cover that is not locally finite
I could not understand that why is not locally finite for example 13.4 can you give me explanation please.
4
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0answers
39 views
Levi-Civita Connection for 2-dimensional Riemannian manifold
I'm trying to show the following. Suppose $(M, g)$ is a $2$-dimensional Riemannian manifold with connection $\nabla$. Suppose also that $\nabla$ is metric compatible, and that length extremizing ...
2
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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 ...
1
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2answers
27 views
Relationship between second order derivatives and cross derivative of smooth surfaces
Probably a silly question, but I wonder if $z=f(x,y)$ is a smooth surface, and the values of its two second order derivatives $\frac{\partial^2f}{\partial x^2}$ and $\frac{\partial^2f}{\partial y^2}$ ...
1
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1answer
39 views
explain $df(tx).x = \sum_{i=1}^n {\partial f\over \partial x_i}(tx)x_i \hspace{1cm} x\in \mathbb R^n$
the question is :
let $U$ be a Neighbourhood of the origine of $R^n$ and :
$x\in U \Rightarrow tx \in U , \forall t\in U $
let f be a numeric function defined in U , and $f(0)= 0$
if we have ...
4
votes
2answers
95 views
Every manifold admits a vector field with only finitely many zeros
Let $M$ be a smooth manifold. I am trying to prove that $M$ admits a vector field with only finitely many zeros.
This will follow if we can find a function $f : M\rightarrow \mathbb R$ such that ...
5
votes
2answers
91 views
Differential Geometry - Computation Help
I'm trying to learn differential geometry through one of MIT's online courses (lecture notes found here: ...
0
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1answer
49 views
Meaning of equation $dx=\sum_{A}\omega_Ae_A$.
I am reading some notes about Riemannian Structures. In definition of moving frame I see blow text and can't understand what $dx$ is:
By a moving frame in $U\subseteq \mathbb{R}^N$ we mean a ...
2
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1answer
46 views
Surface with non-zero mean curvature means orientable
Let $M$ be a surface in $\Bbb R^3$ with non-zero mean curvature for every point. How could I show that this implies that $M$ is orientable? By our definition, orientable means that an unitary, normal ...
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1answer
107 views
I did all explanation. Can you just teach me how to calculate this interior product?
Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball.
Show that an orientation form on $S^n$ is
$w=\sum _{i=1}^{n+1}(-1)^{i-1}x^i dx^1∧...∧dx^i∧...∧dx^{n+1}$
I ...
6
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0answers
47 views
Do balls optimize the boundary area for a fixed volume?
I recently saw this question, asking if the circle is the planar figure which has the least perimeter given the area. As far as I know, this is a classical problem, and the answer is affirmative. I am ...
2
votes
1answer
52 views
Manifolds with boundary and definition
Can you help for understanding this definitions in a good way.What is the my problem is that I can not image this definition and proposition in my mind and also ı dont understand the reason that ı ...
0
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1answer
21 views
coordinate transformation - reparameterization map: polar coordinates
I'm trying to find the reparameterization map $\phi$ (bijective and smooth) for the coordinate transformation of a parameterized curve $\sigma(u,v)$ which is reparameterized from $\sigma(r,\theta)$. ...
0
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0answers
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 ...
