#differentialgeometry — Public Fediverse posts

Frenet–Serret Formula ✍️

It explains how a curve reveals its hidden geometry by tracking the way it bends and twists through space. Imagine tracing the path of a roller coaster, a winding river, or the spiral of a DNA strand. At every point along the path, the curve is constantly changing direction, and the Frenet–Serret formulas provide a precise way to describe that change.

They do this by attaching a moving frame of three special directions to each point on the curve. The first points forward along the path, showing where the curve is heading. The second points inward, toward the direction of bending. The third stands perpendicular to both, capturing how the curve twists out of its plane. Together, they form a local coordinate system that travels with the curve itself.

As you move along the curve, these three directions rotate and evolve. The formulas measure this evolution using two key quantities: curvature and torsion. Curvature tells how sharply the path bends, while torsion tells how strongly it twists into three dimensions. If curvature vanishes, the path becomes straight; if torsion vanishes, the curve lies flat in a plane.

Mathematicians and physicists use the Frenet–Serret formulas to study motion, design smooth paths in engineering, understand particle trajectories, and analyze natural shapes. They transform a simple line into a rich geometric story, revealing exactly how space is being navigated at every step.

#FrenetSerretFormula #DifferentialGeometry #Geometry #Mathematics #Math #PureMathematics #AppliedMathematics #MathematicalPhysics #Physics #STEM #ScienceEducation #MathEducation #Curvature #Torsion #SpaceCurves #VectorCalculus #Calculus #LinearAlgebra #GeometricAnalysis

🔍 Oh, joy! Another 2017 gem where complex mathematical concepts are reduced to "three pictures"—because that's exactly how everyone comprehends Maxwell's equations, right? 🤯 Let's ignore the fact that differential geometry isn't really a bedtime story, but hey, who doesn't love a good abstract with a side of donation plea? 📚💸
https://arxiv.org/abs/1709.08492 #matheducation #complexconcepts #maxwells_equations #differentialgeometry #fundraising #HackerNews #ngated

A pictorial introduction to differential geometry (2017)

https://arxiv.org/abs/1709.08492

#HackerNews #differentialgeometry #pictorialguide #mathematics #arxiv #geometry #education

Today my paper "Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds" got published in the Journal of Geometry and Physics!

The paper defines: A globally hyperbolic spacetime $(X,g)$ is observer-refocusing if there exists a point $p$ and a timelike curve $\gamma$ in $X$ so that all lightrays emitted from $p$ intersect $\gamma$. The paper proves that spacetimes $(X,g)$ with $\dim(X)\geq 3$ which are observer-refocusing with respect to a compact timelike curve have compact Cauchy surfaces with finite fundamental group. This extends known results on strongly refocusing spacetimes, which are spacetimes with points $p,q$ so that all lightrays through $p$ go through $q$. Further, observer-refocusing spacetimes of dimension at least $3$ with an analytic metric are strongly refocusing.

These results lead to immediate corollaries in Riemannian geometry: Let $(M,h)$ be a connected, complete Riemannian manifold and let $x \in M$. We call $(M,h)$ a $Z^x$ manifold if all geodesics starting at $x$ return to $x$. We show that if $\dim(M) \geq 2$ and if the return time of unit-speed geodesics starting at $x$ is uniformly bounded, then $M$ is compact with finite fundamental group. Further, if the metric of a $Z^x$ manifold is analytic, then all unit-speed geodesics starting at $x$ return to $x$ at a common time. This resolves the question "Are all $Z^x$ manifolds $Y^x_l$ manifolds for some $l>0$?" posed in Besse's book "Manifolds all of whose geodesics are closed" affirmatively for analytic manifolds.

https://doi.org/10.1016/j.geomphys.2026.105834
https://arxiv.org/abs/2503.23565

#DifferentialGeometry #MathematicalPhysics #GeneralRelativity

[ Lumo Kaŭstikaĵo ]

Matematika diferenciala geometrio priskribanta la ebenan koverton de kurboj spuritaj de radioj disvastiĝantaj tra manifoldo. 🤓 #nerd

~briletanta~

\eZ

#miksang #dailypic #aphotoaday
#Esperanto #photography #photo
#physics #optics #mathematics #maths
#caustics #differentialgeometry
#manifold #manifolds
#shimmering

Differential Geometry: A First Course in Curves and Surfaces [pdf]

https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf

#HackerNews #DifferentialGeometry #Curves #Surfaces #MathEducation #PDF #Resources

Real-life application of the Theorema Egregium https://en.wikipedia.org/wiki/Theorema_Egregium

#mathematics #DifferentialGeometry

'time is a flat circle'? what are you talking about, all circles are flat

#DifferentialGeometry

Equivalent latitude (Climatology 🌍)

In differential geometry, the equivalent latitude is a Lagrangian coordinate. It is often used in atmospheric science, particularly in the study of stratospheric dynamics. Each isoline in a map of equivalent latitude follows the flow velocity and encloses the same area as the latitude line of equivalent value, hence "equivalent latitude."...

https://en.wikipedia.org/wiki/Equivalent_latitude

#EquivalentLatitude #Climatology #DifferentialGeometry #Equivalence

@lcamtuf

#differentialgeometry #differentialforms

(4/4)

I just really hate how physicist write the cov derivative of the *components* of a tensor (or a section of any vector bundle), while it only make sense if you consider the derivative of the tensor itself.

#DifferentialGeometry #Gravity #GeneralRelativity #Mathematics #Physics

(2/4)
… And for a normal (i.e. \( \mathbb R \)-valued) diff form the cov ext diff \( d_\nabla \) shall be just the same as the normal ext diff \( d \).
This confuses me for a long time.
Until I realised: the eq I wrote above was taken from a #GR textbook, and physicists tend to write every things into coordinates/components/infices format, which brings the confusion.

#DifferentialGeometry #Gravity #GeneralRelativity #Mathematics #Physics

(3/4)
In fact, the tetrad shall be considered as a vector valued 1-form:
\[
theta = \theta_j^i \partial_i \otimes d x^j,
\]
therefore there is no meaning for the cov ext diff for a component of a vector!
One should really consider is the \( d_\nabla \theta\), where the connection is considered on the tangent bundle of the spacetime manifold.

#DifferentialGeometry #Gravity #GeneralRelativity #Mathematics #Physics

(1/4)
I was trying to reason about the (1st) Cartan structure equation
\[ d_\nabla \theta^i = d \theta^i + \Gamma^i_j \wedge \theta^j = 0,\]
where \( d_\nabla \) is the covariant exterial differential, \(\nabla\) is the Levi-Civita connection with connection form \( \Gamma \), and \( \theta \) is an (orthogonal) tetrad.
For me this does not make sense, since \( \theta^i \) is just a normal 1-form, and…

#DifferentialGeometry #Gravity #GeneralRelativity #Mathematics #Physics

Whenever I walk to/from home, I have to walk up/down an inclined street; I noticed that the asphalt floor has different curvatures depending on how near it is of a bend, and I try to find a less steep incline while walking.

This got me inspiration for the few questions below. Any simple explanations, and related links, are welcome.

Given a #differentiable surface within R^3, and two distinct points in it, there are infinitely many differentiable paths from one point to another, remaining on the surface. At each point of the #path, one can find the path's local #curvature. Then:

- Find a path that minimizes the supreme of the curvature. In other words, find the "flattest" path.

- Find a path that minimizes the variation of the curvature. In other words, find a path that "most resembles" a circle arc.

Are these tasks always possible within the given conditions? Are any stronger conditions needed? Are there cases with an #analytic solution, or are they possible only with numerical approximations?

#Analysis #DifferentialGeometry #Calculus #DifferentialEquations #NumericalMethods

From "Fundamentals of Differential Geometry" by S. Lang (1999)

https://link.springer.com/book/10.1007/978-1-4612-0541-8?utm_source=dlvr.it&utm_medium=mastodon

#math #differentialgeometry

Have you seen the latest article from our workshop (non-regular #spacetime #geometry) participants? 👉
Saúl Burgos, José Luis Flores and Jónatan Herrera: The c-completion of Lorentzian metric spaces

#SpacetimeGeometry
#DifferentialGeometry #GeneralRelativity #QuantumCosmology #MathematicalPhysics #Quantum #Cosmology #Relativity #MetricGeometry

@univienna
https://arxiv.org/pdf/2305.02004.pdf

(for the motion picture please visit https://twitter.com/ESIVienna/status/1671167715313344512)

Have you seen the latest article by Brian Allen, a former workshop participant?

#Spacetime #Geometry #SpacetimeGeometry
#DifferentialGeometry #GeneralRelativity #QuantumCosmology #MathematicalPhysics #Quantum #Cosmology #Relativity #MetricGeometry

@univienna

Check out the newest results on #Finsler #Spacetimes by our visitors! 🧐

#DifferentialGeometry #GeneralRelativity #QuantumCosmology #MetricGeometry

@univienna

https://arxiv.org/pdf/2305.04389.pdf

I've worked out that the injectivity radius under the Euclidean metric for the #unitary group U(n) is π and for real and special subgroups O(n), SO(n), and SU(n) is π√2.

This seems like a pretty basic property, but I can't find a single reference that gives the injectivity radii for any of these groups. Anyone know of one?

#DifferentialGeometry #LieGroups #Manifolds

Conjecture: Monoids in the category of smooth manifolds are groups.

Conjecture: A monoid in the category of smooth manifolds with boundary has non-invertible elements iff the boundary is non-empty. In that case the boundary is the maximal subgroup; in particular, the monoid unit lies on the boundary.

#differentialGeometry #showerThought

Spacetime is a four-dimensional topological manifold with a smooth atlas carrying a torsion-free connection compatible with a Lorentzian metric and a time orientation satisfying the Einstein field equations (EFE).
\[\underbrace{(\mathcal{M},\mathcal{O},\mathcal{A},\nabla,\text{g},\mathcal{T})}_{\text{Relativistic spacetime}}\]
#spacetime #space #time #physics #relativity #generalrelativity #specialrelativity #differentialgeometry #manifold #theoreticalphysics #einstein #lorentz

#PhysicsFactlet
Field lines are a convenient way to visualize vector fields, and are defined to be tangent to them at each point.
Due to inertia, field lines do not represent the trajectory that a test mass would follow in a force field.
#VectorFields #DifferentialGeometry #Visualization

Has it ever bothered any of you #math nerds that in #DifferentialGeometry "normal" and "tangent" mean the opposite of their use outside of math?

Rhetorically, a tangent is a digression and an orthogonal concept isn't normal.

#randomthoughts

On December 8, 1865, French mathematician Jacques Salomon Hadamard was born. Hadamard made major contributions in number theory, complex function theory, differential geometry and partial differential equations. Moreover, he is also known for his description of the mathematical though process in his book Psychology of Invention in the Mathematical Field.

http://scihi.org/jacques-hadamard/

#maths #historyofscience #otd #numbertheory #differentialgeometry
#complexfunction