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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Geodesic completeness of a manifold

Let $M\subset \mathbb{R}^3$ be a 3-dimensional submanifold with non empty border $\partial M$. On $\hat M:=M\setminus \partial M$ we put a riemannian metric $g=\frac{dx^2+dy^2+dz^2}{f(x,y,z)}$ where $...
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when do we say a parametrized curve's orientation is consistent with an oriented plane curve

Given an oriented plane curve $C$ and a point $p$ on $C$.Let $A:I\to C$ be a parametrization of a segment of $C$ which contains $p$, where $I \subseteq R$. when do we say $A$ is oriented consistently ...
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A natural Poisson bivector on the tangent bundle?

For a smooth manifold $M$, there is a natural $1$-form $\theta$ on $T^*M$ such that $\Bbb d \theta$ is a symplectic form. Somewhat symetrically, on $TM$ there is a natural tangent field $V$. Is it ...
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1-parametric subgroups of diffeomorphisms induce a complete vector field

I have been working through this book on differential equations and I do not quite understand the justification for one claim. Namely, the author claims that every 1-parameter subgroup $\{\psi_\...
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16 views

Equivalence of definitions of harmonic (or wave) coordinates

In GR, one often uses harmonic (or wave) coordinates to simplify things. Now, one definition involves the coordinates themselves: $$ \Box_g x^{\alpha} = 0 $$ where $ \Box_g = g_{\mu \nu}\nabla^{\mu}...
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1answer
26 views

Integral of $\omega\wedge\overline{\omega}$ on Riemann surface

Let $X$ be a Riemann surface of genus $g$ and $\omega$ a meromorphic 1-form on it. I've read that if $\omega$ has just a simple pole in $x\in X$ (and is holomorphic on $X\setminus\{x\}$) then the ...
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2answers
29 views

Hyperbolic metric geodesically complete

Consider the upper half plane model of the hyperbolic space ($\mathbb{H}$ with the riemannian metric $g=\frac{dx^2+dy^2}{y^2}$). It is known that $(\mathbb{H},g)$ is geodesically complete, which means ...
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1answer
81 views

About codifying length and energy using a one-form

Let $M^n$ be a smooth manifold and $g$ be a Riemannian metric on $M$. Is there $\omega \in \Omega^1(M)$ such that $\int_c \omega = \ell(c)$ for every curve in $M$? In general the answer seems ...
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17 views

Can a section of a signed distance filed uniquely determine this field function?

A Signed distance field function is a field function which tells the minimum distance from any point in space to a specific object. Let $\phi(\vec{x})$ be a signed distance field function, an ...
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are immersions of surfaces in $\mathbb R^3$ dense in all regular maps?

Let $u\in C^\infty(\Omega,\mathbf R^3)$ with $\Omega$ open set in $\mathbf R^2$. Can we find $u_k\in C^\infty(\Omega,\mathbf R^3)$ with $\mathrm{rank}(Du_k(x))=2$ for all $x\in\Omega$ such that $u_k \...
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31 views

Viewing complex projective space as a Grassmannian manifold

Complex projective space $\mathbb{CP}^n$ carries the structure of a complex manifold of dimension $n$, hence has the underlying structure of a real manifold of dimension $2n$. It is the set of complex ...
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32 views

Distributions on submanifolds

I am beginner in differential geometry. I stuck with the concept of distributions(like invariant, anti invariant, slant) on submanifolds. Can you explain what are distributions on submanifolds? If ...
2
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1answer
27 views

“Approximate Isometry” in Riemannian Geometry

I apologize if the notion I'm asking about is well known, I'm no expert in geometry (and I did not find an answer via google). Suppose $(X,g_X)$ and $(Y,g_Y)$ are (smooth) Riemannian manifolds. I'm ...
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26 views

How many kinds of Riemannian metric on $S^n$ up to conformal?

How many kinds of Riemannian metric on $S^n$ up to conformal ? I just happen to get this question,and I think it should has answer.
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1answer
27 views

metric and homotopic maps on a manifold

Let $Y\subset \mathbb{R}^n$ be an embedded manifold without boundary. Prove that there is $\epsilon>0$ with the following property: If $f,g \colon X \rightarrow Y$ are smooth maps defined on a ...
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20 views

Maximal offset distance for a surface

Let $\vec r = \vec r(u, v)$ be a regular (analytic) surface. Now we offsetting this surface to distance $d$ in normal direction; new surface is $\vec r' = \vec r + d\vec n$. New surface $\vec r'$ is ...
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1answer
83 views

Why are geodesically convex sets diffeomorphic to $\Bbb R^n$?

In the construction of a good cover for a manifold $M^n$, Bott & Tu use the fact that each point in $M$ is contained in a geodesically convex set (after picking a Riemannian metric). They then ...
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1answer
8 views

Sectional curvature of 3-hyperbolic space

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{\bigl\langle R(X,Y)X,Y\bigr\rangle}{|X|^2 |Y|^2 - \langle X,Y\...
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3answers
82 views

Why the Jacobian isnt always 1? [on hold]

We have $A=\iint {\rm dx}' {\rm d}y'=\iint G \,{\rm d}x\,{\rm d}y$, where the integral is over a region with area $A$ in the $xy$-plane and $G$ the Jacobian of the coordinate transformation $x\to x'$ ...
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16 views

Effect of gauge transformation on connection 1-form of a principal connection

Let $(P,\pi,M,G)$ be a principal fibration, $A$ a principal connection on $P$ (i.e. $\forall p \in P, T_pP = A_p \oplus V_p$), $\omega$ the connection 1-form of $A$, $f$ a gauge transformation of $P$, ...
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1answer
23 views

function from a genus $2$ surface to $S^1$

Let $f\colon \Sigma \rightarrow S^1$ be a map from a genus $2$ surface to $S^1$. If $y\in S^1$ is a regular value of $f$ and $f^{-1}(y)$ is a nonseparating circle of $\Sigma$. How can I prove that $f$...
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Compactly supported form on $\mathbb{R}^n$

Let $\omega$ be a compactly supported smooth $n$-form on $\mathbb{R}^n$. Show that there exists a compactly supported smooth $(n-1)$-form $\eta$ with $\omega=d\eta$ if and only if $$ \int_{\mathbb{R}^...
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sectional curvature of hyperbolic space

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{<R(X,Y)X,Y>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
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1answer
25 views

Exercise $1.20$ from Montiel and Ros: Curves and Surfaces

Let $\vec{\alpha}:I\longrightarrow \mathbb{R^2}$ be a curve parametrized by arc lenght. If there is a differentiable function $\theta:I\longrightarrow \mathbb{R}$ such that $\theta(s)$ is the angle ...
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1answer
41 views

Tangent vector of a curve

Let $\vec{\sigma}:[a,b]\longrightarrow \mathbb{R}^2$ be a regular and closed curve of class $C^1$, parametrized respect to the arc lenght. Is true that the map $\vec{\sigma}':[a,b]\longrightarrow S^1$ ...
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1answer
45 views

Neglected constant curvature difference surfaces

What are some surfaces where $ \kappa_1-\kappa_2$ is constant? On a sphere where all are umbilical points.. is a special case. For the $ \kappa_1+\kappa_2$ = constant case we have DeLaunay and ...
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1answer
37 views

Construct smooth mapping $f: B^{n + 1} \to S^n$ with two singularities at which $f$ has degree $+/- 1$.

I'm currently working through a paper by Pjotr Hajlasz who wants to show that For smooth manifolds $M,N$, if $\pi_{[p]}(N) \neq 0$ and $1 \leq p < n = \dim M$, then the smooth mappings $C^\...
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41 views

addition of two differential forms with different degrees

Does it make sense to add two differential forms with different degrees like $dx+dx\bigwedge dy$? If yes, what's the arguments of it? I ask this because in text book, the vector space, $\Omega^*(M)$, ...
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Extending vector fields defined on open sets

I'm interested in finding sufficient conditions for when a vector field $X$ defined on an open subset of a smooth manifold $M$ can be extended. It is clear that this can't always be done. For example $...
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How do I find the induced Riemannian metric of a real smooth complete intersection?

If I have a smooth complete intersection of $f_1,\ldots,f_k \in C^\infty(\mathbb{R}^n)$, presented as the vanishing locus $$ f_1 = 0 \text{ } \cdots \text{ } f_k = 0 $$ in $\mathbb{R}^n$, how can I ...
2
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1answer
102 views

Why must a function be independent of coordinates?

What is the motivation for why a function should be independent of coordinates? In the case of a general manifold I kind of get why, since one (usually) defines a function $f$ as a map from the ...
2
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1answer
29 views

About the Fermi charts

In the book Topics in Differential geometry, Peter W. Michor defines the Fermi charts for a Riemannian manifold as follows. Let $(M,g)$ be a Riemannian manifold. For simplicity, I assume that $M$ is ...
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1answer
46 views

Prove that $c:\mathbb{R}\rightarrow\mathbb{R}^2$ with $t\rightarrow (t^2,t^3)$ is not regular.

Prove that $c:\mathbb{R}\rightarrow\mathbb{R}^2$ with $t\rightarrow (t^2,t^3)$ is not regular. Ok so this is probably a well-known problem of Differential Geometry but I have problem understanding it....
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1answer
25 views

Orientation under local diffeomorphism

Given regular surfaces $S_1$ and $S_2$ such that $S_2$ is orientable and a local diffeomorphism $f: S_1 \rightarrow S_2$, then why is $S_1$ orientable? What I think that can be done is to choose an ...
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1answer
19 views

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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1answer
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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? [on hold]

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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1answer
28 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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1answer
28 views

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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1answer
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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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28 views

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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1answer
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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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1answer
44 views

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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1answer
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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 ...