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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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Why does my covariant derivative look different?

It is known for $f$ a smooth function and vector fields $X$, the covariant derivative obeys product rule $$\nabla_Y(fX) = (Yf)\nabla_YX+f\nabla_Y X$$ I just did this calculation and i keep getting $\nabla_Y(fX) = X \nabla_Yf +f\nabla_Y…
Lemon
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Flow of a gradient vector field

I would like to know how the flow of a gradient vector field of a function $f$ (without critical points), on a riemannian manifold, behaves in relation with the level sets. I mean when does it take level sets to level sets?.
Miguel
  • 87
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Path integral of a closed form on $\mathbb{R}^{2}\backslash\{0\}$

Let $$ \omega=\frac{-y\; dx}{x^{2}+y^{2}}+\frac{x\; dy}{x^{2}+y^{2}} \in \Omega^{1}(\mathbb{R}^{2}\backslash\{0\}) $$ I understand that the form $\frac{-y}{x^{2}+y^{2}}dx+\frac{x}{x^{2}+y^{2}}dy$ is closed but not exact on…
user227187
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Try to show $X$ is a smooth manifold

Let $u = (u_1,u_2,u_3)$, $v = (v_1,v_2,v_3)$ and $$X = \{(u,v) \in \mathbb{R^3} \times \mathbb{R^3} \mid u_1^2+u_2^2+u_3^3=1, v_1^2+v_2^2-v_3^2=1, u \cdot v=0 \}$$ Then, is $X$ a smooth manifold? What I have in mind is try to apply regular value…
SamC
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Do Carmo :Show a line of curvature C is a plane curve if osculating plane makes a constant angle

Here's the full problem: Assume that the osculating plane of a line of curvature $C \subset S$, which is nowhere tangent to an asymptotic direction, makes a constant angle with the tangent plane of $S$ along $C$. Prove that $C$ is a plane curve. My…
Nitin
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The differentiability of distance function

Let $M$ be a submanifold of $\mathbb R^n$, then is there an open set $\Omega$ in $\mathbb R^n$ such that function $d(x,M)$ (distance function) is smooth on $\Omega$?
Summer
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Differential and Riemannian structure on the cone

I think the cone (or what is also called the "half cone") is a differential manifold but not a smooth manifold. Can anyone help me understand this the nuts and bolts way? How explicitly can I write down the differential structure on the cone? Also…
Student
  • 2,755
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Show that the map $F(q)=\frac{q}{|q|^2}$ is conformal.

Show that the map $F(q)=\frac{q}{|q|^2}$ is conformal. One way to show that a map is conformal is by looking at the first fundamental form. But since we don't have matrix here, we can recall that $\cos\theta=\frac{X\cdot Y}{|X||Y|}$. So if we can…
3x89g2
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Compactness of a subset of a tangent bundle

Let $(M,g)$ be a Riemannian manifold, $K\subset M$ a compact subset, and $$\widehat{K}=\{v_q\in TM:q\in K,v_q\in T_qM,|v_q|_g\leq 1\}$$ How to show that $\widehat{K}$ is compact? I have tried to modify the proof that the product of two compact…
YYF
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when does the fact that the normal bundle $N(X)$ is trivial imply that the tangent bundle $TX$ is trivial too?

Let $\varphi: X\longrightarrow \mathbb{R}^N$ be an submanifold immersed in $\mathbb{R}^N$, then everybody knows that $T\mathbb{R}^N\vert_X$= T(X)$\oplus N(X)$. It is clear that $T\mathbb{R}^N \vert_X$ is a trivial bundle over $X$. My question is: if…
Coffee
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Show that a helicoid is a regular surface.

Let $S = \{(u\cos v, u\sin v, v): 0
Pom pom
  • 43
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Is parametrized surface a surface?

This is a conceptual question. First, I learned these two definitions during my class: Definition 1: Parametrized surface is a map $q(u,v):U \to \text{R}^3$, where $U$ is open and $U \subset \text{R}^2$. Also, we require that $\frac{\partial…
3x89g2
  • 7,542
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Quotient of $\mathbb{R}$ by $2\pi \mathbb{Z}$

I'd like help for the problem: Let the additive group $2\pi \mathbb{Z}$ act on $\mathbb{R}$ on the right by $x · 2\pi n = x+2\pi n$, where $n$ is an integer. Show that the orbit space $\frac{\mathbb{R}}{2\pi n\mathbb{Z}} $ is a smooth manifold. I…
Jr.
  • 4,026
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Surface of revolution and curvatures

Let $f(x)$ be a smooth function. Consider a surface of revolution, \begin{equation} M(u, v) = (f(v) \cos(u), f(v) \sin(u), v). \end{equation} (a) Calculate coefficients of the first and second fundamental forms for the surface; (b) Calculate…
tellap
  • 161
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3 answers

Gaussian curvature of an ellipsoid proportional to fourth power of the distance of the tangent plane from the center?

Is it true that the Gaussian curvature of an ellipsoid is proportional to fourth power of the distance of the tangent plane from the center? I can verify that it holds at the places where the major axes intersect the surface. (Mathworld has an…
user13618
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