# Tagged Questions

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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### 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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### 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 ...
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### “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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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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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### 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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### 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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### 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? [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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### 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 .
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 \...