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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complex submanifolds in complex euclidean space

Assume all manifolds are without boundaries. In Euclidean space $\mathbb{R}^n$, there are many submanifolds (Whitney Embedding Theorem). In complex Euclidean space $\mathbb{C}^n$, are there any ...
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Ring of smooth functions on a manifold and localization with respect to a multiplicative system

Take $X$ a smooth manifold and $x\in X$. It can be shown that the germ of smooth functions around $x$, $C^\infty(X)_x $ is equal to the algebraic $S^{-1}C^\infty (X)$ where $S$ is the set of smooth ...
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56 views

Classification of Möbius Transformations

We know how to classify the points on a surface,by looking the Gaussian curvature at a point in order to guess the shape of the surface near that point.On the other hand we classify the Möbius ...
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Reversible function

I need help. For which $(r, θ, φ) ∈ \mathbb{R}^3$ is the function $$f(r,\theta,\varphi)=\begin{pmatrix}x(r,\theta,\varphi)\\ y(r,\theta,\varphi)\\z(r,\theta,\varphi)\end{pmatrix}=\begin{pmatrix}r\sin ...
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25 views

the map from the horizontal bundle is a submersion or an immersion

Let $\pi \colon M \to B$ be a riemannian submersion and $g$ the metric on $M$. Then we get the vertical subspace $ \mathcal{V}_x = \ker d_x \pi$ and the horizontal subspace $\mathcal{H}_x= \mathcal{V}...
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38 views

On finding a second countable basis for the tangent bundle $TM$

Let $M$ be a manifold. I want to show that the tangent bundle $TM$ is second countable. I know that for a given chart $(U, \phi)$ on $M$ we have a homeomorphism $D_{\phi}$ between $TU$ and $\phi(U) \...
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3answers
32 views

How do I find the normal vector at point $p$ on a cylinder $x^2+y^2=1$?

How do I find the normal vector at point $p$ on a cylinder $x^2+y^2=1$? I would find this normal vector on point $p$ with any graphic of a function like $(-z_x,-z_y,1)$, but in this case I have no $z$ ...
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1answer
51 views

Pullback of euclidean metric on the disc.

$\newcommand{\Im}{\operatorname{Im}}\newcommand{Re}{\operatorname{Re}}$Consider the biolomorphism $$f : D \to H$$ where $H$ is the complex upper hyperplane $\{\Im(z) > 0\}$ and $D$ is the unic disc,...
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1answer
36 views

Top cohomology of a non-orientable smooth surface with boundary.

I would like to know what the singular relative cohomology $H^2(M,\partial M;\mathbb{Z})$ of a smooth connected surface with boundary $M$ is. In the orientable case I did the following: The zero-th ...
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1answer
69 views

Computing Curvature of a Connection (Dirac Monopole)

I'm trying to verify some computations in a paper I'm reading and am feeling a little lost. In particular I haven't been able to properly compute curvature of a connection acting on a line bundle. ...
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1answer
27 views

Is it true that $(\Phi_{-t})_*w\circ \Phi_t=\sum_{n=0}\frac{t^n}{n!}\mathcal L^n_vw$?

Let $v,w$ be vector fields and let $\Phi_t$ be the flow generated by $v$, that is $\frac{d}{dt}\big|_0\Phi_t(x)=v(x)$. The Lie derivative of $w$ in direction of $v$ is usually defined as $\mathcal ...
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0answers
47 views

What curves will satisfy this very intersting property?

Let $c_1,c_2\subset\mathbb R^2$ be differentiable curves. Given that for any rigid transformation $E$ (i.e. combination of reflections, translations, rotations), if $c_1,E(c_2)$ intersect ...
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1answer
91 views

If X surface and $ E=1+v^2 $, $ F=0 $, $ G=1 $, $ e=0 $, show that $ a(t)=X(uo,vo+t) $ is a straight curve

Let $X : U \to \mathbb{R}^3$ be a regular surface with $E =1 + v^2$ , $F = 0$ , $G=1$ , $e=0$ Show that the curve $ a(t)=X(uo,vo+t)$ (for constant $ (uo,vo) $ at $ U $ and $ t $ belong at $ (-\...
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1answer
325 views

Intersecting geodesics in a positive curvature manifold

Suppose $M$ is a connected, compact orientable 2-dimensional Riemannian manifold, with positive Gaussian curvature. I'd like to show that two non-self-intersecting closed geodesics must intersect each ...
3
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1answer
61 views

Cartan decomposition of SO(2n)

I am trying to understand the Cartan decomposition theory, on the following example : $G=SO(2n)$, and $K=U(n)$, and I'm interested in the manifold $G/K$ (an hermitian symmetric space). 1) How do we ...
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1answer
27 views

Contour integral independant of parametrisation

I have a question about the definition of contour integrals in $\mathbb{C}$. The same question could be applied to line integrals in $\mathbb{R}^n$ though. $\Gamma \subseteq \mathbb{C}$ is called a ...
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Orientability of $m\times n$ matrices with rank $r$

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is non-...
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109 views

Relation between the Hessian and Laplacian

Let $(M^{n},g)$ be a smooth Riemannian manifold of smooth boundary $\partial M$. Assume that Ricci curvature of $M$ is $Ric^{M}\geq0$, and the second fund. form of $\partial M$ is $II\geq c>0$. ...
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34 views

Convex, closed plane curve is a Jordan curve

The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it'...
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Confusion with conclusion to positive mass theorem

I am trying to understand the positive mass theorem as it is presented in the survey paper by Corvino and Pollack http://arxiv.org/abs/1102.5050 I am fundamentally confused by the structure of their ...
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1answer
31 views

Does an unbounded gradient implies a non-vanishing hessian determinant

Let $\Phi:\mathbb{R^{n-1}\to R}$ be a vector function. Suppose that exists a set $S_1\subset \mathbb R^{n-1}$ on which $G=\nabla\Phi$, the gradient of the function is unbounded. Does that imply ...
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45 views

conformal structure of a disc

I wonder if the conformal structure of the unit disc $D^2=\{(x,y):x^2+y^2\leq 1\}$ is unique. More precisely, given a Riemannian metric $g$ on $D^2$, is it always true that $g=e^{2u}g_0$, where $g_0$...
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Sign disagreement in curvature form on $U(1)$-bundle

this is rather trivial question to masters, but as I'm totally disoriented by my computation, so deciding to ask. I'm trying to solve the following exercise, 1.1, showing the curvature 2-form $F(X,Y)=...
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30 views

Second derivative/tangential map on a product manifold

Short version: Given $f: M_1 \times M_2 \longrightarrow N$ what is the second tangential $TTf$ expressed in partial tangentials on $M_1, M_2$? The details: Let $M_1, M_2, N$ be manifolds. The partial ...
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Is the Inverse of a Coordinate System on a Differentiable Manifold Differentiable?

In his book Calculus on Manifolds, Spivak shows that a differentiable k-dimensional manifold in $\mathbb{R}^n$ is a set $M$ such that $\forall x \in M$ there exists an open set $U$ containing $x$, an ...
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1answer
38 views

A Young's inequality used to bound curvature terms

I've been having a look at the Gage and Hamilton's The Heat Equation Shrinking Convex Plane Curves (here). In particular I've been working in the Lemma 4.4.2 and some further results where they find ...
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5answers
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Why is a circle in a plane surrounded by 6 other circles?

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm ...
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1answer
25 views

Apply connections to a gauge transformation?

I'm reading Donaldson's book, Floer homology groups in Yang-Mills theory. On page 82, he considers a trivial bundle $P$ over a $4$-manifold $X$ with tubular ends which is equipped with a connection $...
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1answer
19 views

Derivative of the integral of a pull-back form

Let $\omega$ be a $n$-form on the smooth compact manifold $M$ without boundary. Let $X$ be a smooth vector field on $M$ and $\phi_t$ the associated flow. Let $A(t)=\int_M \phi_t^* \omega$. How can we ...
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What are “generalized bases” really called, and where can I learn more?

(Notation: $f \diamond g$ means the composite $g \circ f$.) The following situation occurs frequently: We have an $\mathbb{R}$-algebra $A$, together with a distinguished set $I$ (the "indexing set"),...
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1answer
48 views

Are there characteristic classes for symplectic vector bundles?

Given a real vector bundle, say the tangent bundle of a manifold, some obstructions to this being the underlying real bundle of a complex bundle come from characteristic classes. These can prove the ...
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Orientability of manifold via covering spaces

Let $f\colon M\rightarrow N$ be a regular covering map between connected differentiable manifolds $M,N$ with $M$ orientable. Prove that $N$ is orientable if and only if every deck transformation ...
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2answers
42 views

what is the curvature in differential geometrical \mathbb{R}^3

If $s$ is the natural parameter, then $x'(s)$ and $x'(s+\Delta s)$ are unit vectors. Therefore the angle $\Delta \varphi$ between them is equal to $$ \Delta\varphi=x''(s)\Delta s+ o(\Delta s).$$ Def: ...
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1answer
49 views

Flow of a vector field, equivalent characterizations

Given a vector field $X$ on the manifold $M$ its flow is, loosely speaking, a map $F= F(t,x)= F^t(x)$, in the variables $(t,x)\in I\times M$, such that the curve $t\mapsto F^t(x_0)$ is the unique ...
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1answer
54 views

Fundamental group and curvature

Is there any paper about the $\pi_1$ group and curvature ? Because how close a curve depends on the curvature near the curve . I think there must have some condition which decide whether there is ...
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1answer
74 views

Why is $T$ in $\mathcal T_2^1(M)$?

Let $M$ be smooth manifold and $\nabla$ an affine connection on $M$. Then the torsion tensor of $\nabla$ is the map $T:\mathcal T(M)\times\mathcal T(M)\to\mathcal T(M)$ and $$T(X,Y)=\nabla_XY-\...
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2answers
41 views

The Lie Algebra of $O(n)$ is the set of $n \times n$ skew-symmetric matrices

I'm trying to show that the Lie Algebra for $O(n)$ is the set of $n \times n$ skew-symmetric matrices. Here is what I have so far. Since $O(n)$ is the union of two disjoint subsets, the matrices with ...
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0answers
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A problem possibly about the Whitney Embedding Theorem.

Problem. (1) Suppose $F:S^1 \to \mathbb{R}^3$ is a smooth immersion (i.e. $dF$ is nowhere zero). Prove that there is a unit vector $\mathbf v$ in $\mathbb{R}^3$ such that $\pi_{\mathbf v} \circ F : ...
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1answer
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Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?

Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...
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1answer
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What's the definition of being zero to infinite order at some point?

I'm reading a paper and it says The differential form $\psi$ is zero to infinite order at all points of $F$.That is,all coefficients of $\psi$ are zero to infinite order at points of $F$.(Here $\psi$...
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Discretizations of Differential, Geometric and Topological Notions

I have noticed a recurring theme in Graph Theory / Theoretical Computer Science (abbreviated GT and TCS throughout this post) in that notions typically belonging to differential calculus / geometry / ...
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Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
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0answers
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Geodesic Formula in terms of First Fundamental Form

I may simply be overwhelmed by all the terms in this question, but I am at a point where I feel stuck: Given a surface $X(u,v)$ with $u=u(t)$ and $v=v(t)$, and $F=0$, find a formula for the geodesic ...
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1answer
25 views

Area form of $M^2 \subseteq \Bbb R^4$: does $(-1)^{i+j-1}\det_{i'j'}(n,\nu) = dx^i \wedge dx^j$?

This is a follow up question from this one. I'm asking separately since one way or another, that one is sort of answered. Now, we know that if $M^{n-1} \subseteq \Bbb R^n$, then the volume element $dM$...
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2answers
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Proving the ratio of curvature and torsion is constant.

This question has been asked slightly differently in a few different forums, but I wanted to discuss my approach and see if I was on the right track: Prove that if the tangent lines of a curve make a ...
4
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1answer
543 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 T_{f(x)}...
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1answer
60 views

Why do those terms vanish if the metric is Hermitian?

On this page, the author says: We now define a Hermitian manifold as a complex manifold where there is a preferred class of coordinate systems in which unmixed components of metric tensor vanish ($...
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1answer
44 views

Reference request : manifolds and transitive lie group actions

I would like to show that any manifold $M$ with a transitive ation from a Lie group $G$ is diffeomorphic to $G/H$ where $H$ is the stabilizer of an element in $M$. Do you know any reference where I ...
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47 views

Geodesics, isometries and connections.

I am trying to prove with the differential definition that isometries preserve geodesics. That is: Let $c(t)$ be a geodesic in a $n$-dimensional semirriemannian manifold $M$ and $g$ an isometry of $M$ ...
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1answer
34 views

Fig 1.8 on page 16 of Guillemin and Pollack's “differential topology”

For fig 1-8 on page 16, there is a sentence explaining why it is not a submanifold: "The trouble arises because the immersion is not one-to-one". I am quite confused because the definition according ...