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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Construct a diffeomorphism $\psi: B_1 \to \epsilon\text{-neighborhood of } K$, where $K$ is a subset of a smooth manifold.

I'm currently working through a paper by Brezis on the topology of Sobolev spaces. Right now I am having trouble understanding the following note made by Brezis. Let $M$ be a compact and smooth ...
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Basic question: Curvature transforms under Complexified Gauge Transformation

Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ is a self adjoint complexified gauge ...
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How to find the curve for which the part of the tangent cut off by the axes is bisected at the point of tangency? [closed]

Find the curve for which the part of the tangent cut off by the axes is bisected at the point of tangency.
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Derivative of Exponential map on manifolds

I'm trying to compute the derivative of the map $f:\Sigma\times [0,\delta)\to M$ given by $$f(p,t)=\exp_p tN(p),$$ in $X\in T_p\Sigma$, where $(M^n,g)$ is a Riemannian manifold, $\Sigma\subset M$ a ...
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29 views

Projective linear special group diffeomorphic to $S^1\times \mathbb{R}^2$

How can I prove that $\mathbb{P}SL_2(\mathbb{R})$ is diffeomorphic to $S^1\times \mathbb{R}^2$? I was thinking about embedding $S^1\subset \mathbb{C}$ as rotations and $\mathbb{R}^2$ as dilatations (...
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Geodesics in geodesic balls

It is well-known that in a geodesic ball centered at $p$, the radial geodesic between $p$ and $q$ is the unique minimizing curve. I'm trying to follow the proof of this given in Cheeger & Ebin (...
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Surface element area from constrains

Consider a surface in $\mathrm{R}^n$ defined by $m$ linear constrains: $$\sum_i c_{ki} x_i = 0$$ We assume that the $m\times n$ matrix $c_{ik}$ is full-rank. Then there exists a linear ...
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Explicit isomorphism $H (E, E^0) \cong H_{cv} (E)$

Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ...
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Integral of solid angle of closed surface from the exterior

Jackson derives Gauss's Law for electrostatics by transforming the surface integral of the electric field due to a single point charge over a closed surface into the integral of the solid angle, ...
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The set of Riemannian metrics on a submanifold of $\mathbb{R}^{n}$

Consider a subset $U \subset \mathbb{R}^{n}$. Clearly, $U$ can be considered as a smooth ($n$-dimensional) submanifold of $\mathbb{R}^{n}$. A Riemannian metric on $U$ is a smooth map $g$ which ...
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Property of geodesic in surface of revolution in $R^3$ [on hold]

It is a question of my homework , I really don't know how to start it .
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Commutators in the context of local Lie groups.

Let $G$ be a local Lie group in the neighbourhood $V \subseteq \mathbb{C}^d$ with identity element denoted by $e \in G$. Also, let $$ t \mapsto f(t) = (f_1(t), \dots, f_d(t)) \quad \forall t \in \...
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Is the set where the exponential map is defined an open subset of $TM$?

Let $M$ be a connected Riemannian manifold. Define $O=\{(p,v) \in TM|\, \,exp_p(v) \text{ is defined} \}$. Is $O$ an open subset of $TM$? I know that for every point in $M$, there is a neighbourhood $...
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Do I have the right idea about affine connections?

On a smooth manifold $M$, a vector field is a smooth map $X : M \to TM$, where $TM$ is the tangent bundle of $M$. If $\chi(M)$ denotes the space of vector fields on $M$, an affine connection $\nabla$ ...
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Differentiation under the integral sign in $R^3$

I'm trying to take derivative from an integral. I know about the Reynolds transport theorem, but I do not know how to obtain the unit normal and the velocity. I'm going to take the derivate from the ...
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Construct a global smooth vector field

Assume the following lemma: Let $K$ be a compact subset of a smooth n-dimensional $\mathbb{R}$-manifold $M$ and $U$ an open subset of $M$ such that $K\subset U$. Then there exists a differentiable ...
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1answer
49 views

Vector fields (on a manifold) and terminology

I read in several books (Do Carmo, Riemannian Geometry or John M. Lee, Smooth manifolds) that a vector field $X$ on a smooth manifold $M$ is a mapping which associates to each point $p \in M$ a ...
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Is $(x,y,\ln(\cos x/\cos y))$ a minimal surface?

Is the surface $(x,y,\ln(\cos x/\cos y))$ minimal? Direct calculation of the first fundamental form seems to get one bogged down in trig functions particularly since it is not diagonal.
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The projection of the curvature vector onto tangent plane on Cone

Draw diagrams for cone ( with cone angle less than $360^{\circ}$) to show that the geodesics (generating ray and the warp around) have a projection of the curvature vector onto the tangent plane that ...
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35 views

Metric and harmonic map (or map between manifolds)

There are many questions about what metric can be placed on a given manifold . For example , place a metric with non-negative curvature. As I know , the Gauss-Bonnet theory is useful in this question....
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Property of Holonomy of a Connection

Let $p:E\longrightarrow B$ be a smooth surjective submersion and suppose $\sigma: p^*(TB)\longrightarrow TE$ is a complete connection. If $\gamma:I\longrightarrow B$ is a path (you may add regularity ...
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Ricci curvature of sum of metrics

Is there an estimate for Ric$(g+h)$ in terms of Ric$(g)$ and Ric$(h)$, where $g,h$ are smooth Riemannian metrics? More specifically can one say that the eigenvalues will decrease (resp. increase) if ...
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Uniqueness of minimizing geodesic $\Rightarrow$ uniqueness of connecting geodesic?

Let $M$ be a complete connected Riemannian manifold. Fix $p \in M$. Assume every point in $M$ has a unique minimizing geodesic connecting it to $p$. Is it true that for every point, the only ...
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Can we lift paths of a Lie group quotient $G\to G/H$?

Question: Let $G$ be a Lie group and $H\subseteq G$ a closed normal subgroup. Let $$\pi:G\to G/H$$ be the quotient map. If $\gamma:[0,1]\to G/H$ is a smooth path, can we find a smooth path $\tilde{...
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A exercise of Riemannian geometry . [closed]

In picture below,I don't know how to start the second question . It is obvious that the isometry of $R^3$ keep the dimension , so there exist such isometry. But seemly, it is too simple . Besides, ...
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How to prove that two compact 1-dim Riemannian manifolds with same length must be isometric?

How to prove that two compact 1-dim Riemannian manifolds with same length must be isometric ? I know the compact 1-dim manifold must be homeomorphism to $S^1$ , but how to do a specific isometric ?
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Self adjoint total covariant derivative

Suppose $V$ is a smooth vector field on a Riemannian manifold $M$ and the total derivative of $V$ is self-adjoint (as an endomorphism of $TM$) i.e. $$\left< \nabla V(X), Y \right> = \left< ...
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Torsion and curvature of a linear connection

Could you help me to solve the following problem ? Let $M$ a parallelizable manifold of dimension $n$, {$E_1$,...,$E_n$} a global frame of $M$. Let $X$,$Y$ a vector fields on $M$ with $Y= \sum_{i=1}^...
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1answer
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How to find the unit normal vector given only the equation of the plane?

In the textbook of Differential Geometry by Do Carmo, there is this example: For a plane $ax+by+cz+d=0$, the unit normal vector is $N=(a,b,c)/\sqrt{a^2+b^2+c^2}$. I am trying to understand how did ...
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Smooth path lifting on $S^1$ [duplicate]

Given a smooth map $f:S^1\to S^1$, there is a smooth map $g:\Bbb R\to\Bbb R$ such that $f(\cos t,\sin t)=(\cos g(t),\sin g(t))$ and $g(2\pi)=g(0)+2\pi q$ for some $q\in\Bbb Z$. I'm fairly sure one ...
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Stokes theorem with divergent form

Let $M$ be a manifold and $S$ a smooth imbedded hypersurface $S \subset M$ that divides $M$ into two disjoint connected components: $M \setminus S \simeq M^1 \cup M^2$. Is Stokes theorem, stated as ...
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1answer
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Irreducible representations of the fundamental group of a closed surface in $SU(2)$

For a compact Lie group $G$, consider the map $f : G^{2n} \to G$ given by $f(A_1, B_1, \ldots, A_n, B_n) = \displaystyle\prod_{i = 1}^{n} A_i B_i A_i^{-1} B_i^{-1}$ A theorem of Goldman (from the '...
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Klein Bottle embedding on $\mathbb{R}^4$ [closed]

Prove that no embedding of a Klein bottle in $\mathbb{R}^4$ can be given by a system of equations $f_1=0, f_2=0$ with independent smooth functions $f_1,f_2$ (i.e. where $df_1,df_2$ are linearly ...
2
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1answer
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Group of deck transformations acts properly discontinuously

Let $M$ be a connected (smooth Riemannian) manifold which admits a universal cover $\tilde{M}$. Let $\Gamma$ be the group of deck transformations on $\tilde{M}$. I want to show that $\Gamma$ acts ...
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Principal null Direction

I need to understand what the principal null dierctions are in mathematics. Physicists define a principal null direction in a spacetime as a null vector which satisfies the Penrose-Debever equation. I ...
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Homology and cohomology of 7-manifold

I have the following problem: Let $M$ be a connected closed $7$-manifold such that $H_1(M,\mathbb{Z}) = 0$, $H_2(M,\mathbb{Z}) = \mathbb{Z}$, $H_3(M,\mathbb{Z}) = \mathbb{Z}/2$. Compute $H_i(M,\...
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+50

Prove that $g$ is a submanifold: $g (t,u,v) = (t^2,u^2,v^2,\sqrt{2}uv, \sqrt{2}tv,\sqrt{2}tu)$

We consider $g : (t,u,v)\in \mathbb{R}^3 \mapsto (t^2,u^2,v^2,\sqrt{2}uv, \sqrt{2}tv,\sqrt{2}tu)\in\mathbb{R}^6$. I have to prove that $g(\mathbb{S}^2)$ is a submanifold of $\mathbb{R}^6$. $dg_{(t,u,...
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Bounding Geodesic Curvature

Let $\Sigma$ be a smooth surface, let $p,q \in \Sigma$ and let $n_{p}$ be the normal at $p$. Suppose that $d(p,q) < c$ for some constant $c$. That is $p$ and $q$ are pretty close to each other on $\...
2
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1answer
44 views

Restriction of a $C^{\infty}$ vector bundle over a regular submanifold.

This question is about the content of page 134 of Tu's An Introduction to Manifolds. A $C^{\infty}$ vector bundle or rank r is a triple $(E,M,\pi)$ consisting of manifolds $E$ and $M$ and a ...
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Symbol of differential operator and change of coordinates

Some time ago I posted the question about the change of coordinates in differential operator. Here is the relevant discussion Symbol of differential operator transforms like a cotangent vector The ...
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How different definitions of connections fit together?

I'm working my way through Ivey and Landsberg, Cartan for Beginners, and I'm working on 2.6.13.2(b). After defining, for $X \in \Gamma(TM)$ a vector field, with section $s:M \rightarrow \mathcal{F}_{...
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Deriving an Expression for the Coordinates of a Partial Hollow Torus as a Function of the Angle

I'm modeling a shape that is best described as a partial, hollow torus. Here's what it looks like: http://i.imgur.com/3h4H5KQ.png In my application, the angle can vary from 0 to 85 degrees. I'm ...
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3answers
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Derivative of projection's norm squared with respect to a matrix

Background: Let $M^{n\times k}(\mathbb{R})$ denote the $n\times k$ matrices with real entries. For any smooth function $f: M^{n\times k}(\mathbb{R}) \to \mathbb{R}$, define the derivative $\frac{\...
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When is the universal cover of a Riemannian manifold complete?

Let $(M,g)$ be a connected Riemannian manifold which admits a universal cover $(\tilde{M}, \tilde{g})$, where $\tilde{g}$ is the Riemannian metric such that the covering is a Riemannian covering. I ...
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1answer
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About a proof concerning the relation of Lusternik-Schnirelman-category and cup length

In the proof of the relation between Lusternik-Schnirelman-category and Cup length (of de Rham Cohomology) for smooth manifolds from this note (theorem 2) the argument goes: Let the given manifold $M$ ...
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How to prove this version of the fundamental theorem of calculus for curves in the closure of a domain

Dear Downvoters: if you leave a comment, you can influence the way this post gets modified, if you don't this post might never satisfy you - even though I keep editing Let $\Omega \subseteq \mathbb{...
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What is $u^{-1}TN$ with $u: M\rightarrow N$ be a smooth map

As picture below, $u\in C^\infty(M,N)$, $(M,g)$ and $(N,h)$ are two smooth Riemannian manifold. I don't know what mean the $\frac{\partial }{\partial y^1} \circ u$ , it is $\frac{\partial u}{\partial ...
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1answer
59 views

Equivalences of the definition of smooth vector fields

Let $M$ be a smooth manifold and $X\colon M \to TM$ a vector field on $M$. I'm having some trouble proving that these assertions are equivalent: (i) $X$ is smooth. (ii) for every chart $(U,\varphi) \...
3
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1answer
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Find surface in $\mathbb{R}^3$ with certain tangent spaces

By Frobenius Theorem, in $\mathbb{R}^3$ there exists a smooth surface whose tangent space is spanned by the vector fields $V(x,y,z)=(x^2+y^2,0,-y)$ and $W(x,y,z)=(0,x^2+y^2,x)$. How can I find this ...
2
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1answer
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Local Properties of Immersions and Submersions

This is from Bredon's Topology and Geometry, pp 83~83. He's definitions of immersion and submersion are the following: Now, in 7.3 Theorem, he explains that (using the implicit function theorem) if ...