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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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$U(1)$-connection

Let $M$ be a smooth manifold. I would like to understand why the moduli space of flat $U(1)$-connections modulo gauge equivalence is the torus $$ H^1(M;\mathbb{R})/H^1(M;\mathbb{Z}). $$ How should I see this?
Terry
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Existence of diffeomorphism so that $\int (\phi^* f) \omega = 0$

Can someone help me with this question? Is a qual exam question and I have no idea how to tackle it. Prove or disprove: Let $f\in \mathcal{C}^\infty(S^n)$ be a smooth function and $x_1, x_2\in S^n$ be two points such that $f(x_1)<0
Morton
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Implicit function theorem

Suppose I have the curve $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ given by $\gamma: t \mapsto (\gamma_1(t),\gamma_2(t)) =(t^2,t)$. If I want to apply the implicit function theorem to this to see if $\gamma_1$ can be expressed in $\gamma_2$ at…
Novo
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closed $1$-form that is exact

Let $\alpha=\sum_{i=1}^n f_idx_i$ be a closed $1$-form defined on all of $\mathbb{R}^n$. Verify that the function $g(\textbf{x})=\sum_{i=1}^nx_i\int_0^1f_i(t\textbf{x})dt$ satisfies $dg=\alpha$. Proof. we must show that $\frac{\partial g}{\partial…
user292200
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Addition of vectors given at different points on a manifold

Let's say I have a set of tangent vectors given at different points ($\vec{v}_i \in T_{p_i}(M)$) on a riemannian manifold with metric and compatible levi civita connection and I like to calculate e.g. the "mean" of all vectors of this set. So I need…
Bort
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Obtaining the Rodrigues formula

On $So(3)$ the algebra of a $3 \times 3$ skew symmetric matrices define Lie bracket $[A,B]=AB-BA$ Consider the exponential map $$EXP: So(3) \to So(3)$$. We have the $So(3)$ matrix $$A=\begin{bmatrix} 0 & -c & b \\c & 0 & -a\\ -b & a &…
Al jabra
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How does one characterize surfaces with constant nonzero Gaussian and mean curvature

I know that for any surface, the Gaussian curvature $K$ and mean curvature $H$ satisfy the inequality $H^2 \geq K$ , and the sphere is a surface where that inequality becomes an equation. Thus, the sphere has both constant Gaussian and mean…
Lothar Spatz
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Why does the curvature approach $\infty$ at cusps?

I found the curvature of the astroid $(\cos^3 t, \sin ^3 t)$ to be: $$\kappa(t) = \frac1{3|\sin t \cos t|}$$ The astroid has $\gamma(\pm \pi/2) = (0, \pm 1)$ and $\gamma(0)$ (resp. $\gamma(\pi)$) $= (\pm 1, 0)$ as cusps. When $t$ approaches any of…
user258700
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Help needed in showing $SU(n)$ is a submanifold of $U(n)$

I have proved that $U(n)$, which is a group of unitary matrices, is a smooth manifold of real Dimension $n^2$. Now I am trying to show that $SU(n)$, which is a group of unitary matrices with determinant 1, is a smooth submanifold of $U(n)$ of real…
user166467
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Definition of Coadjoint representation for Lie algebras

I have trouble understanding the definition of the coadjoint representation of a Lie algebra. Typically you first define a natural pairing between the Lie algebra and Lie coalgebra: \begin{equation} \langle, \rangle : \mathfrak{g}^* \times…
Novo
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confusion on exercises from LEE's Book on Riemannian Manifold

I am reading John Lee's "Riemannian Manifolds an Introduction to Curvature" . At page 15 exercise 2.3 asks to prove that there exist a smooth extension of a function defined on a embedded submanifold. The exercise is wrong for example we can take…
Illuminata
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A $C^{\infty}$ function from $\mathbb{R}^2$ to $\mathbb{R}$

Сould any one help me how to show $C^{\infty}$ function from $\mathbb{R}^2$ to $\mathbb{R}$ can not be injective?
Myshkin
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Vector Field on the $n$-dimensional torus

Give examples of vector fields on the $n$-dimensional torus. What I have done: on $S^1$ it's easy to give one example with perpendicular vectors of length $1$ rotating in one direction, and another example in the other direction. How many…
Maffred
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The regular parametrized curve $\alpha$ has the property that all tangents pass through a fixed point.

There are two questions to this problem: 1) Prove that the curve $\alpha(s)$ is a straight line. 2) Does the conclusion hold if $\alpha$ is not regular, ie. $\alpha'(s)=0$ for some $s\in I$. I proved the first part: Since the lines pass through a…
user53970
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What is a Killing Vector Field?

What is the definition of Killing vector field?. The one my professor told me is : a smooth vector field $V$ on $M$ is called a Killing vector field for $g$ if the flow of $V$ acts by isometries of $g$. So what does it mean by the flow of $V$ acts…
user198206
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