Tagged Questions
3
votes
0answers
41 views
The derivative of a family of flows
If one has a family of flows, can one describe the derivative in the "family" direction?
Specifically, let $M$ be a smooth manifold and let
$X_{s,t}$
be a 2-parameter family of fields on $M$. That ...
1
vote
1answer
67 views
Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k
In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
3
votes
3answers
134 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? ...
1
vote
2answers
46 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 ...
1
vote
1answer
52 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 ...
0
votes
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
1
vote
0answers
44 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 ...
2
votes
1answer
43 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. ...
3
votes
1answer
23 views
Calculating $d\omega$ for $\omega\in\Omega^{k}M$ explicitly for $k=2$
I am trying to explicitly calculate the exterior derivative $d\omega$ for $\omega\in\Omega^{2}M$ for a differentiable oriented manifold $M$.
I know that we can express a differential $k$-form ...
1
vote
3answers
93 views
How to show that an open map $f $ implies the surjectivity of $f'$ in a dense set
Let $f$ be a $C^1$ map from $U\to \mathbb{R}^m$, where $U$ is an open set in $\mathbb{R}^n$, $n\geq m$. Then we know that if $f'$ is surjective everywhere, then $f$ is open.
My question is whether ...
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?
1
vote
0answers
22 views
Gelfand-Leray integral for forms with noncompact support
Let $\omega$ be a smooth $n$-form with compact support on domain $\Omega \subseteq{\mathbb{R}^n}$ and let $f \colon \Omega \to \mathbb{R}$ be a smooth function with nonvanishing differential. Then for ...
1
vote
1answer
54 views
Curvature via hessian in Taylor expansion
In the case of a univariate function, the smaller the second derivative in its Taylor expansion, the smaller is the curvature of the univariate function.
Now, how is the curvature of the function ...
2
votes
2answers
86 views
How does degree theory imply that this mapping $f$ is locally onto?
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth vector field ($\mathcal{C}^1$ mapping). Let $0$ be a critical point of $f$, i.e. $H f(0) = 0$.
Assume that the index of $f$ at $0$ is ...
0
votes
0answers
30 views
Switching differentiation and integration on compact manifold
I'm looking for the theorem stating that differentiation and integration can be switched on compact manifold but I'm not sure there exists such theorem. Can anyone can state the theorem or tell me ...
2
votes
2answers
93 views
Implicit function theorem - how to approach?
I have a question that I have been working on for a while. I was wondering how I should approach the following question:
Are there any points on the graph of the equation
$$x^3+3xy^2+2xy^3=1$$
...
3
votes
0answers
57 views
State of the art of the Implicit Function Theorem
What is the most general form of the Implicit Function Theorem? Quite a general form of this theorem was given by Kumagai (1980): An implicit function theorem. So I am wondering what are the weakest ...
6
votes
4answers
172 views
why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?
Why the following integral means the area of surface $f(x,y)=z$?
$$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
0
votes
1answer
47 views
Prove that a surface of revolution is a 2dimension manifold
I have a question about surface of revolution.
Prove that a surface of revolution is a 2dimension manifold.
1
vote
2answers
69 views
Question on Morse Theory.
I am studying Morse Theory, and I would like to know what a ‘non-degenerate smooth function’ means. Please help. Thanks!
1
vote
2answers
59 views
Smooth maps between Euclidean spaces
There is a question that has bothered me for quite a long time. When we define smoothness of a function(or a map from one space to another), we define it as "has continuous partial derivatives of all ...
1
vote
0answers
22 views
Boundaries- regularity and local parametrization
Suppose we have a bounded domain $\Omega \subset \mathbb{R}^3$ with $C^2$ boundary.Let $x_0 \in \partial \Omega$. We choose a $X_1,x_2,x_3$- coordinate system such that the $x_1,x_2$-plane is ...
1
vote
2answers
66 views
frontier of class $C^{1}$.
Studying the Divergence Theorem (Gauss theorem), found the definition of frontier of class $C^{1}$. Which means? That is, the one which is a set with boundary of class $C^{1}$?
Can give reference ...
3
votes
1answer
69 views
What is the norm of the gradient of $f$ in normal coordinate?
Let $M$ be a Riemannian manifold and $f$ a smooth function on $M$. The Bochner formula proved in Schoen-Yau's book "lectures in Differential Geometry": (prop. 2.2)
$$
\Delta |\nabla f|^2(p)=2\sum ...
2
votes
1answer
87 views
Implicit function theorem-show that in a neighbourhood of the point -can be described by a pair of functions
Let $g_1(x,y_1,y_2)$= $x^2(y_1^2+y_2^2)$-5 and $g_2(x,y_1,y_2)$=$(x-y_2)^2$+$y_1^2$-2. Use implicit function theorem to show that in a neighbourhood of the point x=1, $y_1$=-1, $y_2$=2, the curve of ...
19
votes
0answers
365 views
Ambiguous Curve: can you follow the bicycle?
Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
6
votes
1answer
144 views
Smooth map with surjective Jacobian is open
This is my first question, so I apologize for all the expectations I'm breaking.
I'd like to show that if $U\subset R^n$ is open, $f:U\to R^m$ is smooth, and $J_f(x)$ is surjective (full rank) for ...
0
votes
1answer
26 views
component of a differentiable function differentiable?
in differential geometry we define a function $f:M \rightarrow N$ between differential manifolds to be differentiable, if the function $y \circ f \circ x^{-1}$ (where $y$ and $x$ are appropriate ...
1
vote
2answers
124 views
Are these charts on the circle compatibly oriented?
I've tried a few methods but I can't seem to work this one out.
Consider the charts
$$f(s) = (\cos s, \sin s) \in \mathbb{R}^2$$
for $-\pi < s < \pi$ and
$$g(t)=(\frac{2t}{t^2 + 1}, \frac{t^2 ...
3
votes
1answer
106 views
Relationship beween Ricci curvature and sectional curvature
Let $(M,g)$ be a Riemannian manifold and assume that for all orthonormal $v,z$ the sectional curvatures is bounded from below i.e. $K(v,z) \geq C$, where $C > 0$. Is it in this case possible for ...
1
vote
1answer
59 views
Continuity of the orthogonal projection into tangent space.
Let $\mathcal M \subset \mathbb R^d$ be a smooth manifold, and for each $s \in \mathcal M$ let $T_s[\mathcal M]$ denote the tangent space of $\mathcal M$ at $s$. For each $s \in \mathcal M$ let $P_s$ ...
0
votes
0answers
24 views
Divergent on $M^ n$-Submanifolds of $R^{n+p}$
I was reading a proof (I won't tell by who 'cause I don't if it is true) the author come up with
$\int_M \exp (-|x|^2) Div_M(\nabla _V V)^T=\int_M \exp (-|x|^2) <\nabla _V V,x^T>$
where ...
4
votes
1answer
132 views
Showing a certain operator is trace class.
Let $E \to M$ be a vector bundle over a closed manifold $M$. Suppose $T$ is an endomorphism from $L^2$ sections of $E$ to itself. How does one prove that $T$ is trace class if the image of $T$ is ...
1
vote
1answer
176 views
Geometric meaning of Gram determinant
Let $v_1,v_2$ be vectors in $\mathbb{R}^4$.
Let $M$ be the $2\times 4$ matrix with rows $v_1,v_2$ in this order. The
Gram determinant of $M$ is defined as the determinant of the $2\times 2$ matrix
...
3
votes
2answers
125 views
definition of the exterior derivative
I have a question concerning the definition of $d^*$. It is usually defined to be the (formally) adjoint of $d$? what is the meaning of formally?, is not just the adjoint of $d$? thanks
0
votes
0answers
46 views
Integral transformation
I'm familiar with the transformation theorem in $\mathbb{R}^n$: given $\varphi : \Omega \rightarrow \mathbb{R}^n$ which is a diffeomorphism, $\Omega$ open, then
$$\int_{\varphi(\Omega)} f(y) dy = ...
1
vote
1answer
96 views
How to calculate first variation of functionals defined on curve
I wonder if someone can explain to me how one goes about to calculating the first variation of functions defined on curves. For example, if $C$ is a curve in $\mathbb{R}^3$, and the functional $$F(C) ...
1
vote
0answers
74 views
The Implicit Function Theorem and open sets with regular boundary
Let $\rho :\mathbb{R}^N\longrightarrow\mathbb{R}$ be a continuously differentiable function such that $\rho (x) = 0 \Rightarrow d\rho (x) \not = 0$ for all $x\in\mathbb{R}^N$.
Suppose $\Omega ...
0
votes
1answer
206 views
What does it mean to say a boundary is $C^k$?
I need a explanation on what does it mean to say a boundary is $C^k$. Can anyone help me please.
And also need some explanation on how to straighten boundary ?
1
vote
1answer
54 views
to show $g$ attains maxima and minima
Let $A$ be a symmetric $n\times n$ real matrix and define $G:\mathbb{R}^n\rightarrow \mathbb{R}$ by $G(t)=\langle At,t\rangle$; let $g:S^{n-1}\rightarrow \mathbb{R}$ be the restriction of $G$ to the ...
0
votes
0answers
82 views
Tangent Vectors and Differential 1-forms.
I have this 1-form on $\mathbb R^3$ given by $\omega=dz+\frac{x}{2}dy-\frac{y}{2}dx$. If $p_0=(x_0,y_0,z_0)$ and $\vec v=(u_0,v_0,w_0)$, then find the set of tangent vectors $\vec v_{p_0}$ such that ...
0
votes
0answers
42 views
When to make a substitution in ODE
The setting is on evolving hypersurfaces. So for each time $t$, $\Gamma(t)$ is a hypersurface given by the zero level set of the function $\phi(x,t)$. Consider a ball, then the hypersurface has ...
3
votes
1answer
153 views
Graph and manifold
I needed help to prove the following:
Let $k, n, m$ be elements of the natural numbers and $g : R^m \to R^n$.
Prove that the graph of $g$ is an $m$-manifold of class $C^k$ if and only if $g$ is of ...
3
votes
1answer
301 views
integral of Laplacian of a positive function
I've encountered the following, rather elementary, problem:
$K$ is a compact subset of some 2-dimensional oriented manifold with smooth boundary, $f$ is a positive smooth function on $K$ that ...
2
votes
3answers
89 views
polynomial-torus
I am wondering if it is possible to prove (or come up with an explicit example) that there is a polynomial $f ( x,y,z )$ of degree 8 such that the set $f ( x,y,z )=0$ is a union of two torri? Any ...
2
votes
1answer
77 views
Equality of integrals of differential forms
I have two $(n-1)$-forms $\omega_{1}$ and $\omega_{2}$ on $\mathbb{R}^n$ and a smooth function $g(x) \colon \mathbb{R}^n \to \mathbb{R}$ ($dg$ doesn't vanish anywhere) such that $dg \wedge \omega_1 = ...
4
votes
1answer
129 views
The set of diffeomorphisms preserving some metric.
Let $M$ be a finite-dimensional, smooth manifold. Call a diffeomorphism $f : M \rightarrow M$ diagonalizable if there exists a Riemannian metric $g$ on $M$ such that $f : (M, g) \rightarrow (M, g)$ is ...
1
vote
0answers
52 views
Smooth mapping $v \colon [0,1] \to S^{n-1}$
I have a smooth mapping $v \colon [0,1] \to S^{n-1}$ such that for any $u \in S^{n-1}$ exists $t \in [0,1]: v(t)\cdot u = 0$ and $n \geq 3$. So a have an assumption that such a mapping $v(\cdot)$ ...
4
votes
1answer
96 views
Defining Measures on a Manifold: How To
Everyone:
Given smooth manifolds $M,N$ ($m$- and $n$- manifolds respectively) Sard's
theorem says that for $f:M \to N$ in $C^k$ ; $k \geq 1$, the image of the set of
critical points of $f$ in $M$ ...
2
votes
1answer
137 views
$k$-dimensional area on an $n$-dimensional hypersurface
Let $H_n$ be a $n$-dimensional hypersurface covered by a parametrization $\Phi=\Phi(u_1,\ldots, u_n)$, for $(u_1, \ldots, u_n) \in D$, and let $H_k$ be a k-dimensional hypersurface contained in $H_n$, ...
