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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

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Hopf theorem for non-orientable manifold

I have an exercise which says that :extend the Hopf-Poincare theorem for non-orientable manifold with the indication using the double covering. I have got stuck for long time, so I don't know if somebody can help me. Any answer is greatly…
mapping
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Finding the equation of a curve using binormal vectors

I just started taking Differential Geometry and I'm stuck on this question: "Mark off the line segments of fixed length on the binormals to a simple helix. Find the equation of a curve that is traced by the endpoints of these line segments." I know…
Kelsey
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Geodesics that cross a level set at isolated points

Consider a geodesic complete Riemannian manifold $M_f$ defined by the level set: $ M_f := \{ x \in \mathbb{R}^n | \, f(x)=0 \} $ and a differentiable function $g:M \rightarrow \mathbb{R}$ which defines the level set $N_g := \{x \in M_f | \,…
jaogye
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Inverse image of a regular value is orientable

In Carmo's Differential Geometry of Curves and Surfaces, the book proves that given a differentiable function $f: \mathbb{R}^3 \to \mathbb{R}$ and $a \in \mathbb{R}$ a regular value of $f$, $f^{-1}(a)$ is an orientable surface in $\mathbb{R}^3$. The…
Jeffrey Dawson
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An open subset of a Riemannuan manifold is a Riemannian manifold

The question shows that an open subset of a manifold is a manifold However I think that statement cannot be extended for Riemannian manifolds because the geodesic curve between two points might not be defined, unless the open subset is geodesically…
jaogye
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Is every local diffeomorphic chart a local isometry?

Consider the change of coordinates in $\mathbb{R}^3$ from cartesian $(x,y,z)$ to cylindrical polars $(r,\phi,u)$ i.e. $x = r \, cos \phi$, $y = r \, sin \phi$ and $z = u $. This transformation is obviously an isometry of $\mathbb{R}^3$. Since a…
jaogye
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Showing precisely why sphere in $\Bbb R^n$ has two charts

I'm new to differential geometry, and I haven't seen any concrete examples in it yet, so I do not yet know how to show things there. I've read that the sphere $S^2$, for example, is mapped to $\mathbb{R}^2$ via stereographic projection by removing…
sequence
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Frenet-Serret to show something lies in the plane

Possible Duplicate: Prove that curve with zero torsion is planar How can we use the Frenet-Serret formulas to prove if $\alpha(s)$ is a unit speed curve with $\kappa \neq 0$ and $\tau = 0$, then $\alpha(s)$ lies in a plane. I am thinking maybe to…
mary
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Proving Existence and Uniqueness of Connection 1-Forms

I am trying to solve this exercise from Lee's Riemannian Manifolds, but am getting stuck. The problem is: let $\{E_i\}$ be a local frame with dual coframe $\{ \varphi_i\}$. Show that there exists a unique matrix of $1$-forms $\omega_i^j$ such…
malxmusician212
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Hairy "disk" theorem

Given a disk $D=\{x\in\mathbb{R}^2||x|\leqslant1\}$. There is a continuous tangent vector field $X:D\to\mathbb{R}^2$, which is always pointing towards outside of the disk on $\partial D$. To prove is there exist a zero of $X$ inside $D$. The given…
Upc
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Proof of Kneser's nesting theorem, norm of the evolute of a planar curve with monotone curvature

In these lecture notes http://people.math.gatech.edu/~ghomi/LectureNotes/LectureNotes5U.pdf there is a proof of Kneser's nesting theorem. In one of the steps it is stated that $$\int_{t_0}^{t_1} \Vert \beta'(t)\Vert\,dt = \int_{t_0}^{t_1}…
Rodrigo
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Why is the angle $ OAP=\frac{r_{0}\,\theta}{{r}} $?

I am reading the solution to a problem that involves the figure During the solution the author makes the statement that $ OAP=\frac{r_{0}\,\theta}{{r}} $, where $r_{0}$ is the radius of the big circle and and $r$ is the radius of the small…
gbd
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Square Roots of Hyperbolic Functions

I have the following vector function: $$ r(t)=(t-\sinh t\cosh t)\,\partial_x+2\cosh t\,\partial_y. $$ I computed its velocity to be as such: $$ r'(t)=-2\sinh^2t\,\partial_x+2\sinh t\,\partial_y. $$ Therefore, its speed is as follows: $$ v(t)=2\sinh…
wjmolina
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The index of zero of vector field is well defined

Let $X$ be a vector field on a manifold $M$ of dimension $m$ (compact, connected, oriented) with isolated zero $z$. Let $B$ be a ball around $z$ such that $X$ has no other zeroes in $B$. We define the index of $X$ at $z$ is the degree of the map $f$…
PAM
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Covariant derivative along a curve

Concerning the using covariant derivative along a curve to find its geodesic curvature. Let S be an oriented surface, and $\alpha(u^1, u^2) =(\alpha^1(u^1, u^2), \alpha^2(u^1, u^2))$ a curve in S parametrized by arclength. Then the unit tangent…
Gustav Franklin
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