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 ...
2
votes
0answers
28 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 ...
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votes
0answers
41 views
How prove prove the identity $J_{t}=J\triangledown\cdot V$
prove the identity
$$J_{t}=J\triangledown\cdot V$$
where $J=\begin{vmatrix}
x_{u}&x_{v}\\
y_{u}&y_{v}
\end{vmatrix}$
is the Jacobian of the map $x(u,v;t)\in R^2$ and
$$V(x)=x_{t}$$
3
votes
2answers
46 views
Different types of domains $\Omega \subset \mathbb{R}^n$ in PDEs
In PDEs I often read things like:
Let $\Omega$ be a bounded
Lipschitz or
$C^1$ or
$C^2$ or
$C^\infty$
domain
But I have no clue what this means in real life. I understand ...
1
vote
1answer
28 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 $$A ...
1
vote
0answers
16 views
coordinate transformation of the local pull back of the Maurer Cartan form
This questions asks how to express the pull back of the Mauere Cartan form on a Lie group to a smooth manifold.
The Lie group,$G$, is the structure group of a smooth vector bundle over the ...
1
vote
1answer
34 views
Orientability of $P_{\bf R}T{\bf RP}^{2n}$
I know the following fact :
(1) $ {\bf RP}^{2n}$ is non-orientable.
(2) $ {\bf RP}^{2n-1}$ is orientable.
(3) $P_{\bf R}T{\bf RP}^{2n}$ is orientable.
(4) $P_{\bf R}T{\bf RP}^{2n+1}$ ...
0
votes
1answer
24 views
Question about a specific case of the argument principle for maps of circles.
Problem Statement: Let $f:S^1\rightarrow S^1$ be a smooth map of manifolds where $S^1=\frac{[0,1]}{0~1}$, and let $f'(t)\in \mathbb{R}$ be given by the $df_t[1]_t=f'(t)[1]_{f(t)}$ at each $t\in S^1$. ...
2
votes
2answers
30 views
Extending a smooth map
When can I extend a smooth map $f:\mathbb{R^2}-\lbrace 0 \rbrace \to S^1$ to a smooth map $\tilde{f}:\mathbb{R^2} \to S^1$. For instance, consider $g(x,y)=(x,y)/\sqrt{x^2+y^2}$? Am I able to extend ...
2
votes
1answer
50 views
Gradient in Riemannian manifold
I have a calculation involving a gradient and a parametrization, but I haven't been able to find out the relation between them. Let me explain.
Let $f:X↦R$ be a smooth function and $\mathrm{grad}f\in ...
3
votes
0answers
42 views
Second derivative of a metric in terms of the Riemann curvature tensor.
I can't see how to get the following result. Help would be appreciated!
This question has to do with the Riemann curvature tensor in inertial coordinates.
Such that, if I'm not wrong, (in inertial ...
2
votes
1answer
48 views
Show that there is no surjective smooth function $S^1 \to S^1\times S^1\times S^1$
This is not homework, but a sample test question. The question is:
Show that there is no surjective smooth function $$S^1 \to S^1 \times S^1 \times S^1.$$
Now I can see that, for example ...
0
votes
1answer
23 views
what will be the parameterization of cone
I have question, I need more idea. can any one answer my question I have tried but i didnot get full idea I know this question we have to use parameterization of cone which i donot know in this case ...
2
votes
1answer
24 views
what is the inner product appeared in front of the integral?
Given a compact manifold with a Riemannian metric $g$, we define the total
scalar curvature by
$$E(g)=\int_M RdV$$
Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
1
vote
0answers
36 views
Tangent Vectors in a Surface
As of recent, I've been studying Differential Geometry per the Dover Publication on the subject, and I've ran into a bit of an issue with tangent vectors to a parametric surface $ \mathbf{x}(u^1,u^2) ...
3
votes
1answer
104 views
Diffeomorphism without fixed points [duplicate]
Suppose I have a nowhere-vanishing vector field $X$ on a smooth compact manifold $M$. I am trying to prove that there is some diffeomorphism $f:M \to M$ such that for all $p \in M$, $f(p) \neq p$.
...
