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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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Prove that there exists a unique $ v \in \mathbb{R}^3 $ such that $ u \times v = w $ and $ u \cdot v = 1 $

Let $ u, w \in \mathbb{R}^3$ be such that $ u \neq 0 $ and $ w $ is orthogonal to $ u $. Prove that there exists a unique $ v \in \mathbb{R}^3 $ such that $ u \times v = w $ and $ u \cdot v = 1 $ My attempt: Let $u = (u_1 , u_2 , u_3)$, $w=(w_1 ,…
Curious
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The torsion on the spherical curve

I am currently studying for an exam in differential geometry. There's a problem which I am not able to solve and do not even know where to start. There is the curve β on the spherical surface. which radius is a. If curve $\alpha$ is defined as…
eccentric
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Parameterization of the intersection of two non-parallel planes.

Reading a book on Differential Geometry I saw a statement that says the following: Given two nonparallel planes $ a_{i} x + b_{i} y + c_{i} z + d_{i} = 0 $, $ i = 1, 2 $, their line of intersection may be parametrized as $$ x - x_0 = u_{1} t, y -…
Curious
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Plane Curve Global Geometry / circumference of closed curve

Suppose we know that for every point $c(t)$ on a closed, strictly convex curve $c:I\rightarrow \mathbb{R}^2$, there is a unique point $c(t')$ such that $e_1(t)=-e_1(t')$. $c$ is said to have a constant width if $d(c(t), c(t') )=d$, a…
tcchhlegend
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First fundamental form

Wolfram MathWorld defines a paraboloid and its differential parameters as \begin{align*} P&=\left(\frac{\partial x}{du}\right)^2+\left(\frac{\partial y}{du}\right)^2+\left(\frac{\partial z}{du}\right)^2= \\ &=1+\frac{1}{4u} \\ Q&=\frac{\partial…
user2286339
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Is the exponential map globally continuous?

Question: Given a geodesically complete regular surface $S\in\mathbb{R}^3$, $p\in S$, there is a well-defined exponential map $\exp_{p}:T_p S\rightarrow S$ which is a local diffeomorphism at $0\in T_p S$. It seems that $\exp_p$ must be continuous on…
WLOG
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If $X$ is a vector field and $f\in C^{\infty}(M)$, then is $Xf\in C^{\infty}(M)$?

Let $M$ be a smooth manifold, $X$ be a vector field on $M$ and $f\in C^{\infty}(M)$ be a smooth function on $M$. As obvious as it may sound, by $f\in C^{\infty}(M)$, I'm interpreting this as a map $M\to\mathbb{R}$ that sends any point $p\in M$ to…
Shirish Kulhari
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Lie derivative of a function (of a point) with respect to a vector field

Say $f$ is a function of point on $M$, we define $L_Xf$ to be $\lim_{h\rightarrow 0} \frac{f(\phi_h(p))-f(p)}{h}$, where $\phi_h(p)$ is like (but is not) '$p+h$': moving $p$ in manifold $M$ for a displacement 'proportional' to $h$ along vector field…
Charlie Chang
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Naming of contravariant vector field and covariant vector field

I don’t know why contravariant and covariant vector field are named as such. contravariant literally means going against changing, or changing in the opposite way, covariant literally means changing with something in the same way. For example,…
Charlie Chang
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Schwarz theorem on manifolds

Suppose $M$ is a manifold of class $C^2$ and I have a function $f:M\times M \to M$ which lies in $C^2$. I can define a partial derivative such that I differentiate by one argument, e.g. if $\varphi: \mathbb{R}\to M$ such that $X=[\varphi]\in T_pM$,…
laura_b
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Why does Gauss lemma specify a small sphere?

Shouldn't any sphere on any manifold always be perpendicular to geodesics through its center. This seems obvious to me, since given any function of a distance from the center of a sphere is the equipotential surface, and the flow lines of the…
Kugutsu-o
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Curvature of Fernet curve on a sphere

The question is, how to prove that the curvature of any Frenet curve on a sphere with radius $R$ is bigger or equal to $1/R$. I have managed to prove so far that the Gauss curvature of the sphere $x^2+y^2+z^2=R^2$ is $1/R^2$, but I don't know if…
Despina Plyta
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The gradient in the boundary of a surface

Let the surface in question be the annulus $S := S^1 \times [0,1]$, just as an example. Let $f:S \longrightarrow \mathbb R$ be a $C^\infty$ function. Question: Is there a way to adapt and define the derivative at a point of the boundary of S? (My…
Nelson Schuback
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Components of a tangent space vector along a parametrized curve

I'm studying 'A Most Incomprehensible Thing - Notes towards a very gentle introduction to the mathematics of relativity' by Collier, specifically the section 'More detail - contravariant vectors'. To give some background, I'm aware that basis…
Shirish Kulhari
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Geometric intuition for convexity of hypersurfaces in Riemannian manifolds.

I would like to get a geometric intuition behind convexity of hypersurfaces in Riemannian manifolds: Recall that a hypersurface in some Riemannian manifold is said to be convex, if its second fundamental form is positive definite. Can you tell me,…
nicolas
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