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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144 views

Sard's Theorem Using Integration?

How exactly do we clean up this heuristic proof of Sard's theorem, from Schwarz' `Differential Topology for Physicists' using the Jacobian $J(f)(x)$: "A heuristic justification for Sard's theorem ...
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90 views

Stereographic projection with de Sitter space and hyperbolic plane

How can we do stereographic projection using de Sitter space $\Bbb S^2_1$ and the hyperbolic plane $\Bbb H^2$, in Lorentz-Minkowski space $\Bbb L^3$. For $\Bbb S^2_1$ it is not clear what point ...
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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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1answer
89 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
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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27 views

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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33 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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15 views

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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29 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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21 views

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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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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53 views

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
46 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 ...
0
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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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14 views

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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1answer
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 $g_0$...
0
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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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1answer
53 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 ...
2
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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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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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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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1answer
23 views

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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113 views

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 ...
2
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1answer
41 views

Need help understanding local coordinates of differential forms

I was trying to check a variation of theorem 2.2: I did the first part of the proof using the standard contact structure on $\mathbb R^{2n + 1}$ which worked out as expected. Then I wanted to do it ...
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31 views

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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2answers
48 views

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 ...
0
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0answers
18 views

Definition of complete integrals - existence of envelope?

Let suppose a partial differential equation: $$\Phi(x,y,z,\partial_x z,\partial_y z)=0\qquad (1)$$ In some books I have found the following definition: Let $\Lambda$ and $\Omega$ be two open subsets ...
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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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43 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 $M$ ...
2
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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 ...
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0answers
30 views

Velocity vector of a parametrized curve

In fig. 2.4, at the point of intersection of curve shouldn't we will be having two velocity vectors? How to handle that? Or we will be having only one velocity vector.If that is the case then why is ...
0
votes
1answer
24 views

Curve with arc length have signed curvature k(s)>0?

Let $g:I \to \mathbb{R}^2$ be a curve such that for all $s \in I$, $\|g'(s)\|=1$ and $\kappa_g(s) \neq 0$, where $\kappa_g$ is the signed curvature of $g$. Is $\kappa_g(s) \gt 0$ for all $s \in I$? ...
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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: ...
3
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0answers
50 views

What is the index of a vector field with positive divergence?

Let $v:\Bbb R^n\to\Bbb R^n$ be a smooth vector field with an isolated zero at $z\in\Bbb R^n$. Suppose that $(\operatorname{div}v)(z)>0$. Can we say anything about the index of $v$ at $z$? I know ...
0
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1answer
39 views

Curvature Scalar in Riemannian Space

Suppose that Riklm=a(gilgkm-gimgkl ) on some four dimensional Riemannian space and a is a constant. Question: Show that for the curvature scalar we have R=-12a. What I know from calculating the ...
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34 views

If for each $\alpha, \beta$, the map $\psi_{\beta} \circ F \circ \phi_{\alpha}^{-1}$ is smooth, then $F$ is smooth.

Let $F:M\to N$ be a map. Suppose $\mathcal{A} = \{(U_{\alpha},\phi_{\alpha})\}$ and $\mathcal{B} = \{(V_{\beta},\psi_{\beta})\}$ are smooth atlas for $M$ and $N$ respectively. Suppose that for each $\...
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1answer
24 views

The set of Morse point is open and dense, Exercise 3.3.23 in DoCarmo's Differential Geometry

Let $S\subseteq\mathbb{R}^3$ be a smooth surface, no boundary (not necessarily compact). For any $r\in\mathbb{R}^3\backslash S$, define a smooth function $h_r:S\to\mathbb{R},q\mapsto |q-r|$ Let $p\in ...
2
votes
1answer
42 views

Geodesic curvature of a curve in the hyperbolic plane

Consider the curve $\gamma$ given by $y=b$ in the upper half-plane equipped with the hyperbolic metric $$\dfrac{dx^2+dy^2}{y^2}$$ Calculate the geodesic curvature of $\gamma$. The problem I'm ...
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1answer
21 views

On the zero in the fibre of a vector bundle.

Let $X$ be a differentiable manifold, let $\{U_i \mid i\in I\}$ be an open cover of $X$, let $\{g_{ij}:U_i\cap U_j\to\mathbb{R}^r\}$ be a set of differentiable maps satisfying the cocycle condition, ...
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86 views

Why are symplectic manifolds and Riemannian manifolds so different?

They seem superficially similar in some ways 1) both have a given two tensor on the tangent space 2) if we choose a hamiltonian function for the symplectic manifold then both give a canonical ...
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1answer
19 views

Question in integral curve

Can anybody please help me by explaining why they have evaluated $x_1(t)$ and $x_2(t)$ at $0$. Last second expression where they found $/alpha(t)$.
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1answer
30 views

Simple proof of the existence of lines in the hyperbolic space

Let $\mathbb{H}^n$ be the hyperbolic space defined as warped product: $$ g_{\mathbb{H}^n} = dr^2 + \sinh(r)^2 g_{\mathbb{S}^{n-1}}. $$ What is the easiest way to show that there exist at least one ...
0
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1answer
30 views

Points on the curve

We have to find points on the curve $ax^2+ay^2+2 bxy=c$ (Where c>b>a ) whose distance from origin is minimum . I am not getting any start . I am able to just find that the curve would be hyperbola
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38 views

The graph of any function $f: \mathbb R^n \to \mathbb R$ is a level set for some function $F: \mathbb R^{n+1} \to \mathbb R$

Show that the graph of any function $f: \mathbb R^n \to \mathbb R$ is a level set for some function $F: \mathbb R^{n+1} \to \mathbb R$. Attempt: Define $F: \mathbb R^{n+1} \to \mathbb R$ as $F(x_1,...
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1answer
29 views

The isotropy of the action of $SU(3)$ on $\mathbb CP^2$

Consider the action of $SU(3)$ on the complex projective plane $\mathbb CP^2$. How we can find the isotropy group?
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70 views

What's the point of studying differential geometry? [duplicate]

I've been taking a graduate differential geometry course this semester, and since the beginning I have wondered why one should try to learn that subject. It doesn't mean I don't like it, because I ...
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36 views

Avoiding the spherical polar coordinate singularity on $S^2$ by using a double cover?

Is it possible to avoid the spherical polar coordinate singularity on $S^2$ by taking the coordinates as they originally are on $T^2$, i.e. ranging from $0$ to $2\pi$ mod $2\pi$? How would one ...