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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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For a closed plane curve, showing some inequalities.

I have a problem following : Let $\gamma:[0,T]→\mathbb{R}^2$ be a closed plane curve, i.e., a regular parametrized curve such that $ \gamma$ and all its derivatives agree at 0 and $T$. For convenience of formulation, we assume that $\gamma$ is…
NNNN
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Why is the tangent space to a real projective plane two dimensional?

Let P be the projective plane obtained by identifying antipode points on the unit sphere. How to prove that the tangent space at $q \in P$ to the projective plane P is 2 dimensional? My questions are 1, P is not a submanifold of the Euclidean space…
noot
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Is there geometric interpretation to Skew symmetric coefficient matrix,

We know that the Frenet-Serret equation implies that the coefficient matrix of $\dot t,\dot n,\dot b$ is anti symmetric wrt $t,n,b$. But is there any geometric intuition that immediately gives this result? Thanks!
Golbez
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Surface from metric

I have a metric $g_{i,j}$, in two dimensions for example. How do I find the surface that it represents, in paramatrized form, as a function $z=f(x,y)$, etc. ?
BinaryBurst
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Evaluate the Integral of a 2-form over a Torus

This is a problem from a past qualifying exam that seems a bit too calculation intensive so I was hoping that someone might see a better way to approach this. We are given the 2-form $$\omega = \frac{xdy\wedge dz + y dz\wedge dx + z dx\wedge…
Bohring
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Curvature parametrized by arc length

Suppose $\alpha$ is a curve parametrized by arc length and there is some $s_0$ such that $||\alpha(s)||\le ||\alpha(s_0)||$, $\forall s$ near $s_0$. Show that: $$\kappa(s_0) \ge \frac{1}{||\alpha(s_0)||}.$$ I know the curvature that is…
Lays
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How to induce a connection on endomorphism bundle?

Assume $\nabla:C^\infty(E) \rightarrow C^\infty(T^*X \otimes E)$ is a covariant derivative and $u$ is an element in the endomorphism bundle $End E$. I'm confused why is the induced connection of $\nabla$ on $End E$ is $\nabla^{End(E)}u=[\nabla,u]$.…
Proton
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Simple question on differential forms

Let $\omega = dp_i \wedge dq^i$ be the standard symplectic form on $\mathbb{R}^{2n}(q^i, p_i)$. (We use the Einstein summation convention throughout). Assume $dq^i = h^{ij} dp_j$ on a Lagrangian near $0$ for some function $h^{ij} = h^{ij}(p)$. Then,…
warzasch
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Properties of $\Gamma$ regarding $\operatorname{Hom}$ and $\otimes$.

In Tu's book on differential geometry he defines a connection on a smooth vector bundle $E \to M$ as $\nabla : \mathfrak{X}(M) \times \Gamma(E) \to \Gamma(E)$ so that $\nabla$ is $C^\infty$-linear on $X \in \mathfrak{X}(M)$ and $\Bbb R$-linear on $s…
Jonathan
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planar points and differential geometry

Prove or disprove ; Let $S $ be a surface in $ R^3$. $S $ is a plane iff every point of $S $ is planar point. "All points of plane are planar points" is trivial. But,... the converse is also really true? The definition of planar point ; $p $…
Chris kim
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Is $y=|x|$ a differential manifold?

In the Euclidean space $\mathbb R^2$, the function of absolute values $y=|x|$ determines a $1$-dimentional topological manifold $M$. Define the natural projection map $$p:\mathbb R^2\to \mathbb R,$$ $$(x,y)\mapsto x,$$ let $f:=p|_M$ be the…
Tom
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Differential fails to exist in a certain point in a surface, does this means the surface is not regular?

I am trying to answer the following problem: Show that the two-sheeted cone, with its vertex at the origin, that is, the set $\{(x,y,z)∈R^3 :x^2+y^2−z^2=0\},$ is not a regular surface. I am trying the following: I rewrite it as $(x,y,\pm…
Red Banana
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Pullback of a $1$-form

All: I looked at the list of similar questions, but none seemed to be done explicitly-enough to be helpful; sorry for the repeat, but maybe seeing more examples will be helpful to many. So, I have a differentiable map $f: M \rightarrow S^1 $ , and…
user87588
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Chern's definition of G-structure

Section 4, page 10 of The geometry of G-structures by S. S. Chern, Bull. Amer. Math. Soc. 72(2): 167-219 (March 1966), the definition of G-structure is somewhat vague, and I have have the impression that it is wrong. Let $T$ be an $n$-dimensional…
Arnaud
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Explicitly writing out a differential 2-form

In Tu's An Introduction to Manifolds, one question asks: At each point $p\in \mathbb{R}^3$, define a bilinear function $\omega_p$ on $T_p(\mathbb{R}^3)$ by: …
Ruvi Lecamwasam
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