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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).

32835 questions
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Complete non compact riemannian manifolds and cylinders

Let M be a complete oriented non compact Riemannian manifold with universal covering space (endowed with covering metric) conformally equivalent to the complex plane (with euclidean metric). Then M is locally conformally equivalent to the complex…
user55449
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Angle sum of rays

The sum of angles of a triangle depends on the curvature of the surface and can deviate from $\pi$. What about the sum of angles between successive lines emanating from a given point P? Can it deviate from $2\pi$, depending on the curvature at P? (I…
Hans-Peter Stricker
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Proof that the Gauss Map for a regular embedding $M \subset \mathbb{R}^3$ is a bijection if and only if $M$ is convex

Let $M \subset \mathbb{R}^3$ be a regularly embedded surface. How can I show the Gauss map $n:M \to \mathbf{S}^2$ is bijective if and only if $M$ is convex? In the reverse direction I planned to use the definition of convexity with tangent plane:…
mxnoqwerty
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Möbius band-like strip

A möbius band can be parametrized by the following $ x = (R+r\cos(1/2\theta))\cos(\theta)\\ y = (R+r\cos(1/2\theta))\sin(\theta)\\ z = r\sin(1/2\theta) $ with $R = 1, r \in [0,1], \theta \in [0,2\pi]$ However what is this manifold called? $ x =…
1233023
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unique Ricci flat metric on non-compact Calabi-Yau manifolds

It is known that, for compact Kahler manifolds with vanishing first Chern-class, there is a unique Ricci-flat metric in a given Kahler class. What is known in the non-compact case? Are there certain assumptions you can impose on the asymptotic…
user6013
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topology on $\text{Hom}_{\mathbb R}(E,E')$ for two real vector bundles $E \to M$, $E'\to M$

Let $\pi: E \to M$ and $\pi': E' \to M$ be two real vector bundles and denote the set of all linear maps between the fibres $E_p$ and $E'_p$ by $\text{Hom}_{\mathbb R}(E_p,E'_p)$. I want to show that the disjoint union$$\text{Hom}_{\mathbb R}(E,E')…
Philipp
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How do I calculate, for an ellipsoid of the form

How do I, numerically, inscribe a sphere inside an ellipsoid, $$ ax^{2}+by^{2}+cz^{2}+2fyz+2gzx+2hxy+2px+2qy+2rz+d=0 $$ such that it touches the ellipsoid at a point $(x_1,y_1,z_1)$ on its surface, and has the same curvature as the point? I was…
Ayush Agrawal
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Showing whether a regular curve in R3 with constant positive curvature is always in a sphere

I'm having trouble with showing: Let $\alpha$ be a regular curve in $R^3$ parametrized by arc length with constant positive curvature. Is the curve always in a sphere? I already know that such curve is a planar curve iff it is (a part of) a…
Measurezero
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Decomposition of normal curvature of curve in $\mathbb{R}^3$

Given curve $c$ in $\mathbb{R}^3$, there are $3$ 2-dimensional submanifolds $N_i$ containing $c$ s.t. they are ruled surfaces $$N_i(s,t)=c(t)+s V_i$$ where $\{V_i\}$ is an orthonormal set. Clearly, $N_i$ has a intrinsic metric so that $c$ has a…
HK Lee
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Surfaces which are diffeomorphic

So I just finished my differential geometry exam few days ago, and there is this one bonus problem that bugs me, its a part of a series of questions. BONUS PROBLEM: We give $\Lambda$ is a regular surface, where $(0,0,0) \notin \Lambda.$ Note the…
Aurora Borealis
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Significance of the n-differential forms

I know that it is well defined too say that a differential $k$-form on a manifold M with dimension n is somewhat regarded as a map from the tangent space of the manifold $M$ to $\mathbb{R }$ for each $p\in M,$ then we can define $w$ as the $k$-form…
Aurora Borealis
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Need help with clarifying the wording of the following question concerning manifold.

Let $\ S\ $ be the set defined by the equations $\ {x}^{2}+{y}^{2}+{z}^{4}=3\ $ and $\ {x}^{3} - {y}^{3} + z(1+xy) =2 \ $ Let $\ f(x,y,z)=e^{x+yz} + y{x}^{3}.\ $ Show that, for $\ P=(1,1,1)\ $and some $\epsilon > 0,\ $ $$\ M=S \cap…
Seth
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(1,1) tensor field maps vector to vector

I am studying general relativity by Schutz. I am in curvature chapter, and the following concept I encountered. Say $\vec V$ is vector field, and $x^\alpha$ be a set of coordinates, e.g. in polar case $x^1=r$ and $x^2=\theta$. No we are defining…
user3001408
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Non-Linear Beltrami Vector fields

Consider two concentric toruses, and let $\Sigma$ be the domain interior to the greater torus and exterior to the smaller torus. Is it possible to find a vector field $\mathbf{b}$ satisfying the following conditions: $\mathrm{div} \, \mathbf{b}=0$…
user48900
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Change of metric of embedded surface by normal vector?

Given an embedded surface given by $x_i\rightarrow X^\mu(x)$ where x is 3D and X is 4D. The intrinsic metric is $g_{ab}(x) = \partial_a X^\mu(x) \partial_b X^\mu(x)$. Apply a small shift at each point given by $\delta X^\mu(x) = N^\mu(x)$ where N…
zooby
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