Tagged Questions

1
vote
2answers
58 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 ...
0
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1answer
32 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
0
votes
1answer
39 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.
19
votes
0answers
333 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 ...
5
votes
0answers
152 views

How does one determine $n$-spheres of curvature?

I am aware of circles of curvature and I am simply wondering to what extent does this generalize to $n$-dimensions. Specifically, if some surface in $n$-dimensional space is represented ...
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 ...
3
votes
0answers
58 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 ...
1
vote
0answers
40 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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0answers
21 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
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
0answers
70 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 ...
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 $v(\cdot)$ ...
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 ...
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 ...
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 = ...
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 ...