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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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Show that $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant.

Let $r(s)$ be a curve parametrized by arc length, and $\kappa,\kappa',\tau$ are non-zero. Show that $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant. The teacher gave the hint "center =…
JSCB
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Jet bundle - affine bundle

Suppose $X\rightarrow V$ is a fibre bundle. $X^{(r)}$ denotes the bundle whose fibre at each $x$ is the set of all equivalence classes of $r$-jets. Since $r$-tangency is a finer equivalence relation for larger $r$, it follows that there are…
Karthik C
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A learning question about volume forms

I am learning differential manifold and got a question. How do we calculate the surface area? Or how to calculate the volume of a submanifold? Like for the surface area of $S^n$, if $\phi$ is the embedding map, then it seems…
tomography
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What is trivializing open cover?

Here is where I read "trivializing open cover", but I was not able to find out what it is.
WishingFish
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If $\omega^k$ is exact, is $\omega$ exact?

Let $\omega$ be a differential form on a manifold and suppose $\omega^k$ is exact (but non-zero). Is $\omega$ necessarily exact? Note, the converse is true. If $\omega$ is exact, then $\omega^k$ is exact. To see this, let $\omega = d\alpha$, then…
Michael Albanese
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Exterior Algebra of smooth differential forms

I'm a little bit confused about the exterior algebra of smooth differential forms $\Omega(M)$ on a manifold M. The definition of k-forms is clear to me, but I don't understand how to put them together, s.t. they form $\Omega(M)$ so to speak. Maybe…
Braten
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Normal coordinates vs. Locally flat

If $M$ is a Riemannian manifold the inverse function theorem tells us that for any $p \in M$ the exponential map gives us a nieghborhood $U$ of $p$ and normal coordinates $(x^i)$ in which the components of the metric are $g_{ij}=\delta_{ij}$ and the…
Manuel
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The existence of a local orthonormal frame

Given a Riemannian manifold $(M, g)$ and $p \in M$, is it possible to find a local orthonormal frame about $p$? i.e. in $U \ni p$, there exists $v_1, \cdots, v_n \in TU$ such that $g(v_i, v_j) = \delta_{ij}$. If I want to find a set of orthonormal…
James C
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Change in function's chord length depends on $|s-t|$

I am stuck on this question and any help would go a long way. Show that if $\alpha : [a, b] \rightarrow \mathbb{R}$ is a regular smooth curve and $||α(s) − α(t)||$ depends only on $|s − t|$, then $\alpha$ must be a subset of a circle or a line. I…
PCeltide
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Geodesics which are lines of curvature to surfaces in Euclidean space

How to show that if a curve C in a surface is both a line of curvature and a geodesic, then C is a plane curve. Thanks
user9908
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Integrability of a distribution on $S^{2n-1}$

Let $\theta$ be the restriction of $$ \eta = x^2 dx^1 - x^1 dx^2 + \cdots + x^{2n} dx^{2n-1} - x^{2n-1} dx^{2n} $$ to the unit sphere $S^{2n-1} \subset \mathbb{R}^{2n}$. Since $\theta$ is a nowhere vanishing $1$-form, $\ker \theta$ defines a…
sy32
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how to parameterize a curve f(x,y)?

Given a curve defined as any differentiable function, e.g. $f(x,y)=ax^2+bxy+cy^2+d $, how can I parameterize it into a vector-valued function $c(t) = (x(t), y(t))$? I appreciate any suggestion.
cody
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Gauge transformations in differential forms

I am aware of gauge transformations and covariant derivatives as understood in Quantum Field Theory and I am also familiar with deRham derivative for vector valued differential forms. I thinking of the gauge field A of the gauge group G as a…
Student
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Different forms of 2nd Bianchi's Identity

I have been struggling with relating two different forms of Bianchi's 2nd Identity: $d\Omega = \Omega\wedge\omega-\omega\wedge\Omega$ and $\mathfrak{S}\left\{(\nabla_{Z}R)(X,Y,Z)+R(T(X,Y),Z)W)\right\}=0$ here $R$ and $T$ represent curvature and…
Akhil Nagariya
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For which $a$ is $y^2= x^3 + a$ a submanifold?

I am having a hard time solving the equations to find the $a \in \Bbb R$ for which $$M_a = \left \lbrace {(x,y)\in \mathbb {R}^2} \mid {y^2= x^3 + a}\right \rbrace$$ is a submanifold of $\Bbb R^2$. I defined $F: \Bbb R^2 \to \Bbb R$ with…
mdcq
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