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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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Let G be a topological group that is T2,N2 and locally Euclidean and suppose G admits an open lie subgroup H. Prove that G is Lie

I found this exercise in Tao’s book “Note on Hilbert fifth problem” . Note that if $G$ is connected then $H=G$ since $H$ is also closed and $H\neq\emptyset$ since the neutral element lies in $H$. But if $G$ is not connected I’m a little bit stuck.…
Marco Damele
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Why are the coefficients of a smooth differential form smooth?

Suppose $U$ is an open subset of $\mathbb R^n$, and $\omega:U\to\operatorname{Alt}^k(\mathbb R^n)$ is a smooth differential $k$-form. By $\operatorname{Alt}^k(\mathbb R^n)$ I mean the set of alternating $k$-linear forms, and by "smooth" I mean that…
Joe
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Alternate definition of riemannian manifold

By the Nash embedding theorem, it seems the definition of riemannian smooth manifold is equivalent to the zero locus of some functions $f_1,\dots,f_r : \mathbb{R}^n\to\mathbb{R} $ which are $C^\infty$ and which differentials are zero at no…
V. Semeria
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Proving that in Spherical geometry, Interior angles of triangles don't add up to $180^{\circ}$.

I've gotten a germ of the proof, which involves using the standard metric on the two sphere $ds^2=d\theta^2 + \sin^2{\theta}d\phi^2$. Thing is, how do I continue from here? How do I use this metric to calculate those interior angles regardless of…
Mathipulator
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Curved space described by inverse matrix of another curved space

I was introduced to geometry of curved space in General Relativity but now I am more interested in learning more about it in general.The tensors which describe a curved 3D space can be represented in the form of a matrix A.The eleements of the main…
Cerise
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Elementary Differential Geometry - Reparametrization

I started reading "Elementary Differential Geometry" by Andrew Pressley. I've been confused about Proposition 1.3.6, which reads: A parametrized curve has a unit-speed reparametrization if and only if it is regular. I've been looking over the…
phillip1618
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Hyperbolic manifolds

Does there exist a compact hyperbolic manifold (i.e.,all sectional curvatures are -1) with the same volume as the round-sphere of the same dimension? Does such a manifold exist for any dimension greater than four?
Q-the-curvature
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How to understand Cartan formula $[L_X,i_Y]=i_{[X,Y]}$ geometrically?

Let $M$ be a smooth manifold, $X,Y$ be the vector field over $M$, the Lie derivative $L_X$ is defined by $L_X=d\circ i_X+i_X\circ d$, where $i_X$ means contraction operation of $X$, then by a complicated computation, we can prove the Cartan formula…
Tom
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Curvature Using Circles

Given the equation $(x - h)^2 + (y - k)^2 = r^2$ representing the family of all circles of radius r at the point $(h,k)$ if we try to form the differential equation representing this family we find an equation of the form $$\kappa = \frac{1}{r} =…
bolbteppa
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I've solved this problem, but why is this differentiable?

Let $\alpha:\mathbb R\rightarrow \mathbb R^3$ be a smooth curve (i.e., $\alpha \in C^\infty(\mathbb R)$). Suppose there exists $X_0$ such that for every normal line to $\alpha$, $X_0$ belongs to it. Show that $\alpha$ is part of a…
user79594
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Can a vector field on a non-intersecting smooth curve assign multiple vectors to a point?

I'm reading Semi-Riemannian Geometry by Newman - currently about vector fields on curves. Let $\gamma(t)$ be an injective smooth curve from $(a,b)\subset \mathbb{R}$ to a smooth manifold $M$. I know that for any $t_0\in(a,b)$,…
Shirish Kulhari
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Smooth function motivation in Lee

I was reading a bit in "Introduction to Smooth Manifolds" by John M. Lee for some motivations and came across the following section on page $11$-$12$. "Each point in $M$ is in the domain of a coordinate map $\varphi:U \to \hat{U}.$ A plausible…
user3118
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Basis for the cotangent space $T^\ast_p M$ and the maps $dx^i$

Let $M$ be a manifold and $p \in M$, then we have the tangent space $T_pM$ that has a basis $\left\{ \left(\frac{\partial}{\partial x^i}\right)_p \right \}$. The cotangent space $T^\ast_p M$ has then a basis $\{ (dx^i)_p \}$. I'm trying to figure…
Walker
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Regarding a converse to Hopf's Umlaufsatz

I read in a differential geometry textbook that the total signed curvature of a closed plane curve is an integer multiple of $2\pi$. In that same textbook, I also read about Hopf's Umlaufsatz, which states that the total signed curvature of a simple…
user107952
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Geometric Sig.$\frac{d}{dt}(|\lVert \boldsymbol{r}(t)\rVert )=\frac{\boldsymbol{r}(t).\frac{d}{dt}(\boldsymbol{r}(t))}{\lVert\boldsymbol{r}(t)\rVert}$

Suppose $\boldsymbol{r}\not=0$, what's the geometric derivation/significance of the above, the question is a show that, I could (probably) just use definitions. Instead I thought about when it'd be zero, if the derivative of the magnitude of a…
Alec Teal
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