#differentialgeometry — Public Fediverse posts

[ 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

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

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

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

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 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 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

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

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

#DifferentialGeometry #GeneralRelativity #QuantumCosmology #MetricGeometry

@univienna

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

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

#DifferentialGeometry #GeneralRelativity #QuantumCosmology #MetricGeometry

@univienna

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

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

@charlottekl

Hi Charlotte,

I think a lot about analogies between #Logic and physics, or #SystemsTheory more generally. This led me to see the need for the “missing grape” of #DifferentialLogic analogous to #DifferentialGeometry. A lot of it amounts to differential geometry over \(\mathbb{B} = \mathbb{F}_2 = \mathrm{GF}(2)\) and of course Char 2 makes things a little bit weird and degenerate to a degree but it can be worked out.

@mrdk @narain On the other hand, a parallelogram in the etymological sense, both pair of opposite edges being parallel (their tangents in a parallel transport by other two lines?), is broken as soon as non-zero Riemann curvature is introduced.

In that sense, we might instead want something like a quadrilateral with as many parallel edges as possible. Turns out such shapes have a name.
https://en.m.wikipedia.org/wiki/Levi-Civita_parallelogramoid

#differentialgeometry

Teaching Theorema Egregium in my undergraduate DG class today! My students have done so great this semester I'm really proud of the work they have put in. Our goal is to get to Gauss-Bonnet by the end of the semester. #math #iteachmath #differentialgeometry

#SignRelationalManifolds • 4
• http://inquiryintoinquiry.com/2022/11/05/sign-relational-manifolds-4-2/

Another set of notes I found on this theme strikes me as getting to the point more quickly and though they read a little rough in places I think it may be worth the effort to fill out their general line of approach.

#RepresentationInvariantOntology

• https://web.archive.org/web/20150302021003/http://stderr.org/pipermail/inquiry/2003-April/thread.html#439

• https://web.archive.org/web/20141220180218/http://stderr.org/pipermail/inquiry/2003-April/000439.html

• https://web.archive.org/web/20141220180220/http://stderr.org/pipermail/inquiry/2003-April/000440.html

#Peirce #Semiotics #SignRelations
#Riemann #SergeLang #Manifolds
#DifferentialGeometry #DifferentialLogic

#Peirce #Kant #Riemann
#SignRelationalManifolds
https://inquiryintoinquiry.com/2022/11/01/sign-relational-manifolds-2-2/

Applications of #Manifolds are illustrated by the following excerpts from Doolin & Martin's Introduction to #DifferentialGeometry for Engineers.

https://web.archive.org/web/20110612002240/http://suo.ieee.org/ontology/thrd28.html#04056

Manifolds came up in connection with #Ontology and #Semiotics due to the issues of #Perspectivity, #OntologicalRelativity, and #Interoperability among multiple ontologies. As I see it, those are the very sorts of problems manifolds were invented to handle.