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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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When is a curve parametrizable?

Is there a way in general to tell whether a given curve is parametrizable?
Danny
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Name of isometric invariant in Gauss-Bonnet

Does the tangential rotation term $ \int k_g ds $ of Gauss-Bonnet theorem ( for continuous or discontinuous lines on a surface) have a name or symbol in differential geometry ? The second term $ \int K dA $ integral curvature and the topological…
Narasimham
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Converse of Beltrami-Enneper theorems

To investigate if two of three scalars $ \{k_n =0,\tau_g, K = -1 \} $ are constant, then it follows the third is also a constant. To build a single valid relation among these three scalars viz., geodesics, asymptotic lines on pseudospherical…
Narasimham
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How to prove a surface is smooth

I am given the function $F:\mathbb{R}^3\to\mathbb{R}$ where $F(x,y,z)=(x^2+y^2+z^2-5)^2+16z^2-16$ and then asked to prove that $M:=F^{-1}(0)$ is a smooth surface. Problem is, I wasn't given a definition of a smooth surface in my lecture and can't…
09867
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geometric interpretation of $\vec v \cdot \operatorname{curl} \vec v = 0 $

There is a family of surfaces orthogonal to the vector field $\vec v \in \mathbb R^3$ iff $\vec v \cdot \operatorname{curl} \vec v = 0 $. Now the necessity part is trivial, but the proof of sufficiency I have seen in physics textbooks, e.g. Kestin:…
hyportnex
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Defining the Metric for a Standard Parametrization of a Cylinder

This is very simple. Consider a cylinder in $\mathbb{R}^{3}$. Let the axis of the cylinder coincide with the $z$-axis. Allow the cylinder to be paramterized as follows: \begin{align*} x(\varphi,h) &= a \cos \varphi \\ y(\varphi,h) &= a \sin \varphi…
Stan Shunpike
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verify exercise parallel transport on sphere

I need to solve the following exercise. I wonder whether my solution is correct. Problem: Take a sphere in $\mathbb{R}^3$ centered around the origin of radius $R$. Consider the spherical triangle $ABC$: $A=(0,0,R),\ B=(R,0,0),\ C=(R…
Nadori
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understanding the meaning of formal linear combination and tensor product

I have a question about understanding the meaning of formal linear combination. Let S be a set, the free vector space $\mathbb{R}\langle s\rangle$ on S is defined as the set of all formal linear combination of elements of S with real coefficients.…
user146507
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What are the ricci curvature?

i am trying to find difference among curvatures such as ricci, sectional, scalar, what is the difference between ricci curvature and ricci curvature tensor? i confused. Is there any way to visualized them?
pras
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On definition of tangent vectors of a manifold

I saw the definition of tangent vector on a manifold given as: A tangent vector $v$ at a point $m$ of a smooth $n$-manifold $M$ is a linear derivation of the algebra of germs of functions at $m$. My question is: How can I see (say in a concrete…
self-learner
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Umbilical points of Ellipsoid alternate method

I'm having serious trouble finding the umbilical points of the ellipsoid represented by $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1, \;\;\;a,b,c\neq 0.$$ My first thought was to use the…
Blake
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How to work with Connections

I am currently reading a book which deals with complex manifolds. Since I am fairly new to the topic I don't know exactly the meaning of the followinig: Suppose we have a holomorphic vector bundle $V$ over the manifold $M$ with frames $s_\alpha$…
harlekin
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This equation define a regular surface?

Consider the function: $f(x,y,z)=xyz^2$ Its gradient is $\nabla f=(yz^2, xz^2, 2xyz)$ then the critical points are all in the sets $\{(x,y,0): x,y\in \mathbb{R}\}, \{(0,0,z): z\in \mathbb{R}\}$. My question is: The set defined by $xyz^2=0$ is a…
EQJ
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closed but not exact

I saw several times that $\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy$ is closed but not exact. Closed, is obvious but I can't prove non exactness, can one please help me ? My attempt, let $f\in \omega^{0}(U)$ and…
dragoboy
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exercise on surfaces and geodesics

Maybe someone can verify my answers. The problem is as follows: Consider a line $l$ in $\mathbb{R}^3$ and rotate it around the $Oz$ axis. Denote by $A$ the set of all such obtained surfaces. Question 1: Write their parametrizations. My answer: Let…
Nadori
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