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 ...
0
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
27 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$$
Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
1
vote
0answers
36 views
Tangent Vectors in a Surface
As of recent, I've been studying Differential Geometry per the Dover Publication on the subject, and I've ran into a bit of an issue with tangent vectors to a parametric surface $ \mathbf{x}(u^1,u^2) ...
3
votes
1answer
104 views
Diffeomorphism without fixed points [duplicate]
Suppose I have a nowhere-vanishing vector field $X$ on a smooth compact manifold $M$. I am trying to prove that there is some diffeomorphism $f:M \to M$ such that for all $p \in M$, $f(p) \neq p$.
...
1
vote
1answer
21 views
Is a parameterization defined to be surjective and/or injective?
A parameterization is a mapping used in differential geometry for describing a manifold, and in statistics for describing a family of distributions, and may be used for other applications I don't know ...
3
votes
1answer
29 views
Given a local diffeomorphism $f: N \to M$ with $M$ orientable, then $N$ is orientable.
Given a local diffeomorphism $f: N \to M$ with $M$ orientable. Why is $N$ orientable? My professor wrote this in class without giving a proof and said "you should try to prove this for fun :)". I ...
1
vote
1answer
34 views
geometrically finite hyperbolic surface of infinite volume
I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a
"geometrically finite hyperbolic surface of infinite volume"
is mentioned frequently and I am ...
1
vote
1answer
31 views
Is this distribution involutive?
For two days I've been trying to show the following: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and consider the smooth distribution $$F=\{F_p=DR_p(e)\mathfrak{h}; p\in G\},$$ where ...
0
votes
0answers
46 views
Show that geodesics in the plane are straight lines use $X(u,v)=(u\cos(v),u\sin(v))$ as our parameterization?
I have question
Q
Show that geodesics in the plane are straight lines. Use $X(u,v)=(u\cos(v),u\sin(v))$
as the parameterization of the plane.
I hope someone can answer.
Thanks
please at ...
1
vote
1answer
30 views
Meaning of modulo diffeomorphism
I faced this sentence:
we consider the space of Riemannian metrics modulo diffeomorphism and scaling.
Can anyone explain to me what is the meaning of modulo diffeomorphism and scaling?
Thanks!
1
vote
1answer
34 views
Holonomy of the sphere
I saw an example in which the holonomy of $\mathbb{S}^n$ with the standard metric is calculated. I'm just starting to get familiar with holonomy groups and I wanted to know what could one do by ...
2
votes
0answers
47 views
“Product” bundle notation.
Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively.
Then there is an induced ...
1
vote
1answer
54 views
Question about lie bracket..
Let $G$ be a Lie group with Lie algebras $\mathfrak{g}$ and let $\mathfrak{h}\subseteq \mathfrak{g}$ be a Lie subalgebra. Write $F_p=DR_p(e)\mathfrak{h}$, $p\in G$, where $R_p:G\rightarrow G$ given by ...
0
votes
1answer
45 views
what is the right circular of cone and what is the right circular of cylinder
I have some questions.
1)what is the parametrization of cone and
what is the parametrization of cylinder?
2)
what is the right circular of cone
and what is the right circular of cylinder?
I ...
0
votes
1answer
37 views
Find the differential equations that are satisfied by geodesics on the torus with parametrization given
I have question
Find the differential equations that are satisfied by geodesics on the torus with parametrization given
$X(u,v)=((R+rcos(u))cos(v),(R+rcos(u))sin(v),rsin(u))$?
I hope someone can ...
2
votes
1answer
66 views
Trouble understanding differential forms. A basic question: what does $w \times dw$ mean?
After reading [1] and [2] I (kind of) understand what differential forms are, but I am still having trouble understanding the following argument from [3,Lem.4.2]:
Let $\mathbb{T^3_n}$ be the ...
1
vote
1answer
42 views
Poisson bracket of coordinates
I just derived that in local coordinates (it suffices to centre) around $0$, that
$$\{f,g\}(x)=\sum_{i,j}\{x^i,x^j\}\frac{\partial f}{\partial x^i}\frac{\partial g}{\partial x^j}$$
only using the ...
1
vote
1answer
25 views
Calculating Principal curves
I have been given a surface patch, $X(u,v)$, and I have calculated its unit normal, coefficiants of its first and second fundamental form and found its principal curvatures. Now it's asking me to find ...
1
vote
1answer
45 views
Motivation for the study of the Chern connection
Given a Hermitian metric $H$ over a holomorphic vector bundle $E$ with holomorphic structure $\overline{\partial}$, there exists a unique connection $\nabla$ (named afer Chern) satisying the following ...
2
votes
1answer
60 views
Computing $n$-th external power of standard simplectic form
I need some help:
Define a 2-form on $R^n$ by $\omega=dx_1\wedge dx_2+dx_3\wedge dx_4+...+dx_{2n-1}\wedge dx_{2n}$. How to compute $\omega^n:=\omega\wedge\omega\wedge\ldots\wedge\omega$?
2
votes
1answer
34 views
Computing the unit normal vector - Simplifying help
I have a surface
$$X(u,v) = \left(3uv^2 - u^3 - \frac{u}{3}, 3u^2v - v^3 - \frac{v}{3}, 2uv \right), $$
and the cross product
$$(X_u \times X_v) = \left(3(u^2 + v^2) \frac{1}{3} \right) \cdot ...
2
votes
3answers
45 views
symmetric positive definite matrices
Why must a symmetric positive definite matrix must be invertible? I'm reading a proof of the Levi-Civita theorem in differential geometry but the author states this without proof and I haven't been ...
3
votes
0answers
25 views
+50
Triangulation of a manifold adapted to a submanifold
I am not extremely proficient in topology, and am concerned with the following question :
Given a compact manifold $X$ and a submanifold $Y \subset X$, is it always possible to find a triangulation ...
2
votes
1answer
25 views
Geometric intuition for convexity of hypersurfaces in Riemannian manifolds.
I would like to get a geometric intuition behind convexity of hypersurfaces in Riemannian manifolds: Recall that a hypersurface in some Riemannian manifold is said to be convex, if its second ...
1
vote
1answer
49 views
Equality involving Lie Brackets
I have a question concerning Lie brackets: Consider the Lie bracket $$[, ]:\mathfrak{g}\times \mathfrak{g}\rightarrow \mathfrak{g},$$ where $\mathfrak{g}=T_eG$ is the Lie algebra of a Lie group $G$. ...
2
votes
0answers
37 views
Tubular neighborhood with an additional projection
Let $i\colon L\to M$ be a submanifold inclusion. The tubular neighborhood theorem says that there is a tubular neighborhood of $i(L)$ in $M$ diffeomorphic to the normal bundle of $L$ in $M$, denoted ...
1
vote
1answer
48 views
Two results on the mean curvature of hypersurfaces
I am a physicist, now I consider a physically meaningful $N-1$ dimensional hypersurface $M^{N-1}$ embedding in the flat Euclidean space $R^{N}$. We have an explicit form of the hypersurface in the ...
-1
votes
0answers
46 views
question about geometry
Let $y(t)$ be a regular curve and its curvature $\kappa(t) > 0$ for all $t$. Then $\kappa(t)$
is a smooth function of $t$. Show that it is no longer true without the assumption $\kappa(t) > 0$.
...
-6
votes
0answers
36 views
Find approximate value using diffrentials [closed]
Find approximate value of sin 45° using differentials.
Be detailed about the answer because I don't have time to read the chapter and have a test in 30 minutes.
0
votes
0answers
44 views
Show a cone isn't a regular surface
I know that a cone isn't a regular surface because I can't construct a chart with cts partial derivatives at its tip. But can anyone show me this last step rigorously? Why would any chart for the tip ...
0
votes
0answers
39 views
Computation of a pullback of a two form
If we have a Lagrangian immersion from $C^{2}$ to $C^{4}$ defined like this
\begin{align}
\notag
\phi : (x,y,u,v) \to (x, y, u, v, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, ...
-1
votes
1answer
46 views
Enneper's surface: differential geometry
Let $c \neq 0$ denote a real number. A surface patch is given as follows:
$$ \alpha_c(u,v) =( \frac{u}{c^2} - \frac{u^3}{3} + uv^2,\frac{v}{c^2} - \frac{v^3}{3} + vu^2,\frac{u^2-v^2}{c} ) $$
where ...
1
vote
0answers
27 views
Do sections defined in different patches give the same element in an associated bundle?
we can read here in p. 10 (penultimate equation) that the sections on associated bundles (not necessarly vector bundle) are defined by functions $f:P\longrightarrow F$ that must satisfy the ...
1
vote
1answer
43 views
Why do manifolds with negative sectional curvature not have conjugate points?
I'm trying to understand why manifolds with negative sectional curvature not have conjugate points. In fact for me it is sufficient to understand it for surfaces, but of course i'd be interested in ...
2
votes
0answers
22 views
Are geodesic flows on surfaces with negative curvature Anosov?
I'm just going through the original book by Anosov, where he tries to proof this result.
I don't quite understand it.
So let $\phi_t:TM\rightarrow TM$ be a geodesic flow on a compact surface $M$ of ...
1
vote
0answers
16 views
Bott connection
Can anyone help me showing the following: Let $E$ be a smooth vector bundle over $M$ and $F\subseteq TM$ a smooth distribution. A $F$-connection is a $\mathbb R$-bilinear aplication ...
1
vote
1answer
21 views
Positive curvature on holomorphic vector bundles
There must be a mistake in my understanding the definition of positivity for the curvature. Let me summarize:
Let $ (L,\nabla,h) \rightarrow M $ be a hermitian hol line bundle with Chern connection. ...
2
votes
1answer
32 views
Prove that a tensor field is of type (1,2)
Let $J\in\operatorname{end}(TM)=\Gamma(TM\otimes T^*M)$ with $J^2=-\operatorname{id}$ and for $X,Y\in TM$, let
$$N(X,Y):=[JX,JY]-J\big([JX,Y]-[X,JY]\big)-[X,Y].$$
Prove that $N$ is a tensor field of ...
1
vote
1answer
28 views
Distribution and Tangent Bundle
Let $F=\{F_p; p\in M\}\subseteq TM$ be a rank $k$ smooth distribution. Can anyone explain-me what is the set $$\displaystyle\nu(F)=\frac{T_pM}{F_p}.$$
