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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$M=\{(x,y,z) \in \mathbb{R}^3 : xyz=C\}$ is a manifold $\Rightarrow C \neq 0$

I'm having some troubles on showing that $M=\{(x,y,z) \in \mathbb{R}^3 : xyz=C\}$ is a manifold $\Leftrightarrow C \neq 0$ I have already proved $\Leftarrow$ but I can't see how to prove ...
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85 views

What means that two manifolds have “the same topology”?

I know this is a very basic question, but let me be more specific. Suppose that, for definiteness, $M$ and $N$ are differentiable manifolds. What means that they have the same topology? Does this mean ...
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180 views

Hyperbolic geometry when the curvature is constant and negative but not -1

Help I am getting completely confused Hyperbolic geometry is the geometry of surfaces of a constant negative Gaussian curvature, in most formula's it is almost assumed this constant negative ...
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58 views

$GL(n,K)$ is open in $M(n,K)$

I want to prove, that $GL(n,K)$ is open in $M(n,K)$, where ($K=\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. For the prove I don't want to use that the determinant is continous. Alternativly I assume a ...
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32 views

Basis for curvillinear coordinate systems.

I am reading through Geometry of Physics - Frankel and in the preface of the latest edition, Frankel defines a curve $C_i$ through a point $p$ parametrized by $u^j=constant$, $j\neq i$, with ...
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76 views

Poincaré–Bendixson theorem

Does someone know a good reference for a proof of the Poincaré–Bendixson theorem using the language of vector fields?
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98 views

Preservation of the curvature tensor implies preservation of the connection?

For every connection on a smooth manifold there is a corresponding curvature tensor. Any diffeomorphism $\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ which preserves the connection (in the sense of ...
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57 views

Does this set of symmetric matrices form a smooth manifold?

Let $A$ be a real symmetric $n \times n$ matrix. Let $1 \leq i < j \leq n$ and $1 \leq k < \ell \leq n$. Let $A'$ be the submatrix of $A$ consisting of rows $i,...,j$ and columns $k,...,\ell$. ...
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99 views

Pre-requisites for studying differential geometry?

I am an 3rd year undergrad interested in mathematics.i had read h.graham flengg (from geometry to topology).i found this field interesting i now want to read further,so i want to ask these questions ...
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Finding Gauss curvature of surface

Consider the surface $S=F(\mathbb{R}^2)$ where $F:\mathbb{R}^2 \to \mathbb{R}^3$ is defined by $$(r, \varphi) \mapsto ( r \cos \varphi, r \sin \varphi, \varphi).$$ I would like to find the Gauss ...
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57 views

Applications of the Picard theorem for ODE [closed]

The Picard theorem I am considereing is the one that states, under suitable hypothesis, that an ordinary differential equation has a unique solution in a suitable interval $I\subset\mathbb{R}$. Does ...
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45 views

Derivative of a function - how to compute for those examples

I'm taking a Diferential Manifolds course and I don't understand how to compute $DF_a$ in order to apply the following theorem: Let $F:U \rightarrow \mathbb{R}^m$ be a $C^\infty$ function on an ...
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1answer
36 views

A question on diffeomorphims and their intuitive description

I am relatively new to the concept of differential geometry and my approach is from a physics background (hoping to understand general relativity at a deeper level). I have read up on the notion of ...
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1answer
122 views

curvature form of the unit sphere

Trying to compute explicitly the curvature form of the unit sphere and got the following result: The parametrization of the unit 2-sphere $ S^2$ is well known: $ n=(\sin\theta\cos\phi, ...
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74 views

Maximization: Volume of paraboloid within cone?

Given a right circular cone with the line of symmetry along $x=0$, and the base along $y=0$, how can I find the maximum volume paraboloid (parabola revolved around the y-axis) inscribed within the ...
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1answer
60 views

(1,s) tensor fields

I'm having a hard time understanding what's going on with tensor fields. I understand that $A$ is a smooth covariant tensor field of order $s$ (or a $(0,s)$ tensor field) on a smooth manifold $M$ if ...
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71 views

confusion about basic Kahler geometry

I am really struggling to understand the basics of Kahler geometry and hope someone can give me some guidance. Suppose we have a complex manifold with some complex structure $J$ and let $g$ be a ...
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34 views

computing the metric on a complex manifold

Let $M^n$ be a complex manifold with coordinates $z_1,\dots, z_n$ where $z_k=x_k+\sqrt{-1}y_k$. Let $$\frac{\partial}{\partial z_k}=\frac{\partial}{\partial x_k}-\sqrt{-1}\frac{\partial}{\partial y_i} ...
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81 views

Coding for a calculation in differential geometry using Maple

I am beginner in maple. And my field is Differential geometry. I've learnt lie brackets using maple help. But I am testing this calculation through maple. I have these vector fields $e1=z^2\ast ...
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79 views

Induced Lie Algebra Representation, Left invariant vector fields and more…

The following is an excerpt from a proof in John Lee's Introduction to Smooth Manifolds I am struggling to understand. I would appreciate if someone was able to help me with whatever it is I am ...
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2answers
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Difference between second order derivative and curvature.

I am studying space curves and I found the following equation for the curvature of $f(x)$ when $f(x)$ is a plane curve. $\displaystyle\kappa=\frac{|f''(x)|}{(1+(f'(x))^2)^\frac{3}{2}}$ I always ...
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78 views

Direction of outward normal differential $\hat n\ ds$

In vector calculus, why is the outward normal differential $\hat n\ ds$ around a closed curve $dy\ \hat i - dx\ \hat j$? Why isn't it $-dy\ \hat i + dx\ \hat j$ or something else?
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60 views

Regular Surface Patches

Which of the following are regular surface patches? Let $u,v\in \mathbb{R}$. $(i)$ $σ(u, v)=(u, v, uv)$ $(ii)$ $σ(u, v)=(u, v^2, v^3)$ $(iii)$ $σ(u, v)=(u + u^2, v, v^2)$ I'm not sure how to show ...
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37 views

The Jacobian of the exponential along a geodesic

I am reading a paper that uses but does not define the following concept: what is understood by "the Jacobian of the exponential map along a geodesic (beetween two points)"? Is this only defined for ...
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35 views

determinant of metric on complexified vector space

Let $M$ be a complex manifold of dimension $n$ with Hermitian metric $g$. Extend $g$ to $TM^{\mathbb{C}}$ linearly. I believe that at each $p$ we can say a basis for $T_pM$ as a real vector space is ...
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46 views

Half-space is not a manifold

We define the half-space $H^n$ as the set containing all tupels $(a_1,\ldots,a_n)$ such that all $a_i\geq 0$. I know that this isn't a manifold - intuitively this is clear - but how can I formally ...
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1answer
81 views

Proving one version of equivariant formality

Let $G$ be a compact, connected Lie group acting smoothly on a compact, connected and oriented smooth manifold $M$. We denote by $H_G^*(M)$ the corresponding equivariant cohomology. We have a ...
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1answer
62 views

How to check if a given curve represents a Circle

I have the following parametric curve equation $$ y(t) = (4/5 \cos t, 1- \sin t, -3/5 \cos t) $$ and I am stuck how to find the curve equation as now it has x,y and z, while I have only dealt with x ...
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2answers
234 views

First Cohomology Group

Is it true that the first cohomology group of a differentiable manifold with finite fundamental group is trivial? If so, could you explain why? Thanks very much
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98 views

Closed 1-forms on Simply Connected Manifold

Is it true that closed 1-forms on a simply connected differentiable manifold are exact. If so, could you explain why? Thanks very much
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Introduction to differential equations for pure mathematicians

Is there a good reference for learning about differential equations for people who are mainly interested in the theoretical tools (especially in differential geometry/topology) that use them? I ...
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1answer
124 views

kernel of the exponential map is isomorphic to the singular homology group

Let $G$ be an algebraic torus or an abelian variety over the complex numbers. Then $G(\mathbb{C})$ is a complex Lie group. Is it true that we have the following exact sequence ? $ 0 \to ...
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Prove that $(f\circ \alpha)''$ is not a tangent vector, where $\alpha$ is a diferentiable curve on a manifold.

Let $(M,\Psi)$ be a manifold with a $C^k$ atlas, and let $\alpha: I\subset \mathbb{R}\mapsto M$ be a $C^k$ curve in $M$ with $\alpha(0)=p\in M$. The curve $\alpha$ defines a tangent vector at $p$: ...
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When is the composition of multiple flows the identity?

Suppose I have some number of vector fields $a,b,\ldots,k$, and I denote flow along $a$ for time $t$ by $a_t$, etc. When is it the case that for all points $x$, $a_tb_tc_t\ldots k_tx = x$? In the ...
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1answer
84 views

Canonical isomorphism from $I_p/I_p^2$ to cotangent space

Sorry if the title is confusing, I don't know if the terminology is standard. For my homework this week I have to prove the following: Let $M$ be a smooth manifold and let $p \in M$. Let $I_p$ denote ...
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Lie algebras of GL(n,R) and differentials

This question comes from a proof in John Lee's Introduction to Smooth Manifolds, page 194. I am questioning a line in the proof of the following proposition: The composition of the maps ...
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1answer
197 views

Differential equation of space curve with given curvature and torsion

What is the differential equation of a 3D space curve whose varying curvature and torsion are given as functions of arc length? (upto rotation and translation of Euclidean motions?) EDIT1: ...
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1answer
48 views

Minkowski metric on a surface

Do closed surfaces admit a metric with lorentzian signature? Any reference?
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1answer
63 views

Find the equation of the tangent plane of each of the following surface patches at the indicated points.

Find the equation of the tangent plane of each of the following surface patches at the indicated points: $$σ(r,θ)=(r\cosh(θ),r\sinh(θ),r^2), (1, 0, 1).$$ I'm not sure what to do, any hints are ...
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0answers
52 views

Left invariant vector field question

This question is an exercise from Jeff Lee's Manifolds and Differential Geometry Let $\tilde{X}$ be the left invariant vector field corresponding to $X\in L(V,V)$ (That is, $\tilde{X}_g=(dL_g)_e X$). ...
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29 views

Lie Groups map question

This is a question about Exercise 5.59 in Jeff Lee's Manifolds and Differential Geometry. He writes For a fixed $A\in GL(V)$, the map $L_A:GL(V)\rightarrow GL(V)$ given by $A\mapsto A\circ B$ has ...
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1answer
79 views

Circle determined by three points on a curve tends to the osculating circle

I am stuck on problem 3.3.2 of Differential Geometry of Curves and Surfaces by Banchoff and Lovett. The problem is: Let $\vec{x}(s) \colon I \to \mathbb{R}^2$ be the parametrization by arc length of ...
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1answer
52 views

Proving that on a Lie group $G$ the space of left-invariant vector fields is isomorphic to $T_e G$

Let $G$ be a Lie Group. Given a vector $v\in T_e G$, we define the left-invariant vector field $L^v$ on $G$ by $$L^v(g)=L^v|_g=(dL_g)_e v.$$ I want to show that $v\mapsto L^v$ is a linear ...
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Key differences between almost complex manifolds and complex manifolds

I know the technical difference between an almost complex manifold and a complex manifold, namely in the former the almost complex structure $J$ may not be integrable while in the later it is. ...
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1answer
92 views

Tensor calculus on the frame bundle

Let $M$ be a manifold and let $g$ be a tensor on it, say for example a metric $g\in\Gamma(T^{\ast}M\otimes T^{\ast}M)$. I know how to perform any computation on $g$. For instance, taking its ...
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1answer
68 views

Making sense of polar coordinates transformation on the derivatives

I would like to make sense of the transformation of the differentials in polar coordinates (to fix the ideas). To be more precise, the "right" way to find the transform for the differential and the ...
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1answer
95 views

The smooth Nullstellensatz

Let $n$ be a positive integer, let $f_1, \ldots, f_r : \mathbb{R}^n \to \mathbb{R}$ be smooth functions, let $Z_i = f_i^{-1} \{ 0 \} \subseteq \mathbb{R}^n$, and suppose $Z_1 \cap \cdots \cap Z_r = ...
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1answer
36 views

Prove that the set of all vector fields $V(S^1)$ is a free $C^{\infty}(S^1)$-module

I need to prove that the set of all vector fields, $X:S^1\to TS^1$ name it: $V(S^1)$, is a free $C^{\infty}(S^1)$-module. So i need a basis $\frac {d}{dx_1},...,\frac {d}{dx_n}$ for $V(S^1)$.It's easy ...
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1answer
78 views

Principal directions bissect the asymptotic directions

How can one prove that at a hyperbolic point, the principal directions bissect the asymptotic directions?
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39 views

Is a closed simple curve of the plane entirely determined by the points of extremal or stationary curvature?

The title is rather explicit: consider a $1$-periodic smooth map $f:[0,1)\to \mathbb{R}^{2}$ injective on $[0,1)$ and let $C_f$ be the image of $[0,1]$ by $f$. Let $s$ be the abscissae on the curve ...