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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241 views
Notation to work with vector-valued differential forms
What it the standard notation used while working with vector-valued differential forms?
I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
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1answer
179 views
Difference between pushforward and differential
The pushforward of a map $F:M \to N$ at a point $P \in M$ is defined as $F_*:T_P(M) \to T_{F(P)}(N)$ where
$$(F_*X)(f) = X(f \circ F)$$
where $X \in T_P(M)$.
The differential of a function $f$ ...
5
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1answer
112 views
Basis of cotangent space
The derivative of a map $F$ between manifolds $M$ and $N$ is defined by
$$F_*X(f)= X(f \circ F)$$
where $X \in T_P(M)$, the tanget space at the point $P$.
We know that ...
2
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1answer
233 views
Osculating plane and unit speed curve
If $\alpha(s)$ is a unit speed curve with $k\ne0$, how can we show that the equation of the osculating plane through $\alpha(0)$ is $[x-\alpha(0),\alpha'(0),\alpha''(0)] = 0$. (I mean 3 equal bars for ...
5
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2answers
115 views
Trivilisations of Vector Bundles
Let $\pi : E \to X$ be a smooth rank $k$ vector bundle on a manifold $X$ (I don't think my question depends upon the stipulations on the bundle, but I've just chosen smooth in case I'm incorrect). By ...
2
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2answers
91 views
Equivalence of two definitions of differentiablitity on Regular Surfaces
When dealing with differentiable surfaces one defines a function $f:S\rightarrow \mathbb{R}$ as being differentiable if its expression in local coordinates is differentiable. But one could also define ...
5
votes
3answers
472 views
*understanding* covariance vs. contravariance & raising / lowering
There are lots of articles, all over the place about the distinction between covariant vectors and contravariant vectors - after struggling through many of them, I think I'm starting to get the idea. ...
2
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1answer
96 views
Interpretation of Multilinear maps as tensors
Let M be a smooth manifold and $C^{\infty}(M)$ the smooth functions on it.
Some authors are calling $C^{\infty}(M)$-multilinear mappings: $T:\mathcal{X}(M)^s\rightarrow\mathcal{X}$ Tensor fields of ...
4
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1answer
82 views
germ finitely determined
Does anyone know any result on finitely determined germs to help me prove that the germ $f(x,y)=x^3+ xy^3$ is $4$- determined? I tried using the definition of germs finitely determined, which is:$f: ...
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0answers
96 views
Maple example about Ricci flow collapses the sphere manifold or Einstein manifold to a point in finite time
how to use maple code to demonstrate Ricci flow collapses the sphere manifold or Einstein manifold with positive curvature to a point in finite time
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1answer
53 views
Continuity of the lengths of geodesics on a closed manifold
Let $M$ be a compact Riemannian manifold and let $SM$ be its sphere bundle,
$$SM = \{(x,\xi) \in TM : \|\xi\| = 1\}.$$
There is a well-defined function $\ell : SM \rightarrow [0,\infty)$ defined by ...
1
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1answer
73 views
Parametrization of a solid
Find a parametrization $\sigma : I \subseteq \mathbb{R}^3 \rightarrow \mathbb{R}^3$, with $I$ a parallelepiped, of $\lbrace (x,y,z) \in \mathbb{R}^3 : |z| \leq 4x^2 + 9y^2 \leq 1 \rbrace $.
5
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1answer
146 views
Abstract manifolds or do we need an ambient space?
I'm currently studying on J.Lee's "Introduction to smooth manifolds", but several other sources I consulted present the same line of thought.
The most natural description of the $n$-dimensional ...
2
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1answer
327 views
How to evaluate the derivatives of matrix inverse?
Cliff Taubes wrote in his differential geometry book that:
We now calculate the directional derivatives of the map $$M\rightarrow M^{-1}$$ Let $\alpha\in M(n,\mathbb{R})$ denote any given matrix. ...
2
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1answer
53 views
domain of surface of revolution
Let $0<b<a,(u,v) \in \mathbb{R} \times \mathbb{R}$. Then the map $g(u,v):=((a+b\cos u)\cos v,(a+b\cos u)\sin v,b\sin u)$ defines a torus.
I wonder for $g$ to be a surface does it really need ...
2
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0answers
39 views
laplacian for functions problem with the integral on manifolds
I'm following the proof of the local expression for the Laplacian on a compact manifold and I'm having problems understanding how the integral on a manifold translates into an integral in $R^n$, in ...
4
votes
1answer
100 views
$U(1)$-connection
Let $M$ be a smooth manifold. I would like to understand why the moduli space of flat $U(1)$-connections modulo gauge equivalence is the torus
$$
H^1(M;\mathbb{R})/H^1(M;\mathbb{Z}).
$$
How should I ...
4
votes
2answers
143 views
orthonormal vector fields
In a Riemannian manifold $(M^n,g)$, is it true that given a point $p$ we can find $n$ vector fields $X_i$ on a neighborhood $U$ of $p$ such that $g(X_i,X_j)(x)=\delta_{ij}$ with $x \in U$?
It seems ...
4
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2answers
213 views
Morse function with indices of only $0$ and $n$
Q1: If a Morse function on a smooth closed $n$-manifold $X$ has critical points of only index $0$ and $n$, does it follow that $X\approx \mathbb{S}^n\coprod\ldots\coprod\mathbb{S}^n$?
I think the ...
1
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0answers
50 views
Homogeneity lemma
I am studying Homegeneity lemma. I am not understanding the following paragraph:
Given any fixed unit vector $c \in S^n$, consider the differential equations
$\frac{dx_i}{dt} = c ...
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0answers
155 views
an injective immersion between two compact manifold of same dimension
$f:M\rightarrow N$ be a injective immersion, where $M$ and $N$ are same dimensional manifold with out boundary, we need to show $f$ is a covering map.
what I tried is, $df_x:T_x(M)\rightarrow ...
3
votes
1answer
87 views
Tensor calculation on mean curvature flow
I have two questions about tensor calculation.
First question : In the book, Lectures on mean curvature flows written by Xi-Ping Zhu, there exists the equaility $g^{mn} \nabla_m \nabla_n h_{ij} = ...
3
votes
3answers
126 views
Definition of vector bundle
Everywhere i see definition of vector bundle as triple $(E, p, B)$, $B$ and $E$ are manifold and local trivialization condition holds. For example see the definition here. .
Local trivialization ...
5
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1answer
168 views
Group action on manifold
I met a question as follows:
Suppose $G$ is a finite group acting freely on a manifold $M$. Show the following: 1) $M/G$ is a manifold. 2) $H^i_{dR}(M/G)=H^i_{dR}(M)^G$. 3) if $N$ is a compact ...
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2answers
68 views
$P^1$ not a regular level surface of a $C^1$ function on $P^2$
I'm working through the first chapter of Morris Hirsch's "Differential Topology". On Chapter 1, section 3 exercise 11, I encountered the following question.
"Regarding $S^1$ as the equator of $S^2$, ...
5
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0answers
363 views
What is magical about Cartan's magic formula?
Why is Cartan's magic formula
$$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$
called "magic"?
Should it be considered a highly surprising result? Does it "magically" prove several other ...
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0answers
172 views
Measure theoretic definition of curl
Is there a good measure theoretic definition of curl?
To give an idea of the sort of equation that I'm looking for, here's now I define grad and div. For the gradient, say we are given a Fréchet ...
3
votes
2answers
178 views
Software for differential geometry
What is the software for computations with tensor fields? I'm interested in computations with vector-valued differential forms on Riemannien manifolds. Wedge product, exterior derivative, likely Hodge ...
3
votes
0answers
200 views
Reconstructing a curve from its curvature
I'm with a problem in an exercise form Do Carmo's Differential Geometry of Curves and Surfaces; it is number 9 section 1.6.
I have a differentiable real function $k(s)$, $s\in I$, and I need to show ...
3
votes
4answers
239 views
global section vector bundle
do non-zero global section always exist in a manifold $M$? If $M$ is compact I think they do because taking a partition of unity $\rho_{\alpha}$ subordinated to a finite covering, and defining local ...
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66 views
Isoperimetric Area Function
I'm reading the definition of isoperimetric function area and appeared the following notation:
$Ł^{n}_{g}(\Omega)$, where n is dimension of Riemannian manifold $(M,g)$.
What is the meaning of ...
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2answers
370 views
$\{(x,y)\!\in\!\mathbb{B}^n; -\varepsilon\leq-\|x\|^2\!+\!\|y\|^2\leq\varepsilon\}\approx\mathbb{B}^k\!\times\!\mathbb{B}^{n-k}$
The question is motivated by the notion of handle attachment, Morse theory, critical points of index $k$, Morse lemma, sublevel sets, etc.
For $0\!\leq\!k\!\leq\!n$ and ...
4
votes
0answers
77 views
Variants of isotopy extensions
I am interested in slight variations of the usual isotopy extension theorems.
In short, my question is the following : Can one extend isotopies of $C \subseteq M$, where $C$ is compact and $M$ is a ...
4
votes
0answers
181 views
Morse theory and homology of an algebraic surface (example)
Let $T_n$ denote the $n$-th Chebyshev polynomial and define $f_n(x,y,z)\!:=\!T_n(x)\!+\!T_n(y)\!+\!T_n(z)$ and $$Z_n:=\mathcal{Z}(f_n) \subseteq \mathbb{R}^3,$$
the Bachoff-Chmutov surface, where in ...
4
votes
1answer
62 views
Is there a standard name for a category all of whose contravariant hom functors are sheaves?
What prompted this question is the definition of a
pseudogroup in nlab:
Given a X a topological space. Then a pseudogroup is a subgroupoid of the groupoid of transitions between open sets in X, ...
4
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0answers
209 views
Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?
Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem".
I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
2
votes
1answer
92 views
Showing: point of polytope which maximizes the minimum distance to a vertex is a barycentre?
Let $T_1$ and $T_2$ be two regular $(n-1)$-dimensional simplices with vertices $$(t,0,\ldots,0), (0,t,\ldots, 0),\ldots, (0, 0, \ldots, t),$$ and $$(t-n+1,1,\ldots, 1), (1, t-n+1, \ldots, 1), \ldots, ...
5
votes
1answer
134 views
Morse functions are dense in $\mathcal{C}^\infty(X,\mathbb{R})$.
In Shastri's Elements of Differential Topology, p.210-211, there is written:
Why do we get a Morse function $f_u$ on $X$? We know that for any $f\!\in\!\mathcal{C}^\infty(X,\mathbb{R})$, there is ...
5
votes
1answer
176 views
Pullback of differential form on the double covering
On a double covering there is a differential form $\omega$ arises by the pullback of a differential form under the projection iff it is the pullback of $\omega$ under the map $i$, where $i$ is the map ...
1
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2answers
103 views
Does an orientable subbundle of an orientable vector bundle always have a orientable complement?
If I have an orientable vector bundle $E$ and a subbundle $F$ on a manifold $M$, where both the bundles are orientable, does $F$ have a complement in $E$ which is also orientable? Does it have a ...
4
votes
1answer
102 views
Symplectic Chart
I was reading the article "Symplectic structures on Banach manifolds" by Alan Weinstein. In this article there is one theorem, which is as following:
If $B$ is a zero neighborhood in Banach space. ...
4
votes
1answer
200 views
The connection in terms of local trivialization
I am having trouble to understand how one may write down the connection in terms of local trivialization of the vector bundle.
Assume $\pi: E\rightarrow X$ is a vector bundle with rank $n$. A ...
5
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0answers
233 views
A fiber bundle over Euclidean space is trivial.
What's the easiest way to see this? The only thing I could think to do was try to patch together trivializations. I couldn't find a way to make that work. Thank you!
edit: For the record, here's why ...
3
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2answers
194 views
Is there such a thing as discrete Riemannian geometry?
General relativity expresses gravity as a curvature in space time, created by the stress energy tensor.
$$T_{\mu\nu} \approx R_{\mu\nu} - \frac{R}{2} g_{\mu\nu}$$
Given I put the fact that energy is ...
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0answers
103 views
Closest point projection of manifolds in Banach spaces
Suppose that $Y$ is an infinite dimensional Banach space, with an embedded finite dimensional compact submanifold $X$. It is well known (cfr Lang's Differential and Riemannian Manifolds, Thm. 5.1) ...
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1answer
106 views
distance between a polytope point and a polytope vertex
How to find distance in between any polytope point to the closest vertex of the polytope (the verteces of the polytope are known)?
How to find a distance from the farest polytope point to the closest ...
5
votes
2answers
161 views
What are the subobjects of a manifold?
Categorically a subobject of an object $a$ of some category $A$ is an object $a'$ with a monic morphism to $a$, ie $a'\to a$, upto isomorphism.
When $A$ is either a Topological or Differential ...
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0answers
61 views
Neighborhood Retraction of Boundary
Here is the problem: If $M$ is a manifold with boundary, then find a retraction $r:U \rightarrow \partial M$ where $U$ is a neighborhood of $\partial M$.
I realize that the collar neighborhood ...
2
votes
2answers
69 views
How to find the maximal integral submanifold in a concrete case?
Take $M=\mathbb{R}^3$ be a smooth manifold. Consider a distribution
$\Delta_{(x,y,z)} = Span\{y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y},
z\frac{\partial}{\partial x} - ...
0
votes
1answer
70 views
extension of convex function
Let $\varphi$ be a nonnegative function defined on a domain $\omega\subset R^n$ such that $\varphi$ is $C^2$ and convex on $\omega$. Does $\varphi$ admit a $C^2-$ extension on $\Omega\supset ...
