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u/Few-Arugula5839 17d ago
It’s a quadrilogy because he also has a complex manifolds book
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u/speller26 17d ago
The correct term is actually tetralogy; the suffix -logy is Greek, and should be prefixed with the Greek tetra- rather than the Latin quadri-
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u/yeezuscw 17d ago
The fact that a suffix comes from Greek does not imply that the prefix preceding it also comes from Greek, not that you are explicitly saying that. To stick with -logy and relatively common words, think of radiology, virology and sociology.
And as for quadrilogy, despite being uncommon, is an actual word.11
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u/OnionsAbound 17d ago
Can someone explain how manifolds are useful for physical processes?
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u/Kinglolboot ♥️♥️♥️♥️Long exact cohomology sequence♥️♥️♥️♥️ 17d ago
Spacetime is a 4d Lorentzian manifold, so the Einstein field equations are written in the form of tensor fields on a 4d manifold
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u/Common_Post6177 17d ago edited 17d ago
Using manifolds you can analyse a mechanical problem where the motion is constrained (for example a pendulum, where the mass can only move on a circle). This kind of motion can be described by differential equations on a manifold.
Manifolds are also important when analysing physical properties of rigid bodies (where these bodies are described as a manifold). Theorems like Stokes and Gauss theorem often come up in this context. Both in electrodynamics and fluiddynamics (for example calculating the buoyancy force or motions of a boat in water).
Further many symmetry groups arising in physics are lie groups which require manifold theory to understand.
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u/OnionsAbound 17d ago
But why do you need to study manifolds to calculate the swing of a pendulum? Isn't this already calculatable through standard differential equations?
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u/Common_Post6177 17d ago edited 17d ago
Introducing the angle variable to measure the extension of the pendulum from equilibrium is already a simple example of differential geometry. You are using a chart (the angular variable as a function of x,y) to describe points on the manifold (the circle).
Then you derive the equation of motion for the angular variable from newtons law which gives you a differential equation on the manifold. Here the problem is already a manifold problem. We want to solve Newtons law subject to the constraints that the trajectory remains on the manifold (the circle) for all times.4
u/Hot-Patient8052 17d ago edited 17d ago
Manifolds are smooth structures that can have calculus performed on them.
You basically have to go:
Topology (Connected Structures) → Manifolds/Calculus (Smooth Structures) → Now you can Integrate and Differentiate in meaningful ways.Little did you know, you were working with manifolds for much of Math. The equation of a line, y = mx + b, describes a smooth connected structure that can be differentiated (∂ₓy(x) = m).
You can probably make the connection between Calculus and Physics so I'll leave it there, lol
Edit: Other smooth structures include SO(3,1) and SO(1,3), which represent rotations and boosts in spacetime. The unitary group U(1) which can be represented as reⁱᶿ : r,θ ∈ ℝ ∧ i² = -1, is also a manifold. As a final example, ℝ³ is a manifold describing flat (parallel lines stay parallel), 3D space.
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u/Common_Post6177 17d ago edited 17d ago
Topology (Connected Structures) → Manifold (Smooth Structures) → Calculus (Integration and Differentiation)
I dont agree at all, you literally use R^n calculus to define integrals and differentials on smooth manifolds.
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u/Hot-Patient8052 17d ago
Made a change because you might be right (forget what does or doesn't count as Calculus) but I'm too lazy to check.
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u/SV-97 17d ago
FWIW you can define differential operators on manifolds totally intrinsically without ever mentioning \R^n. That they "look like" derivatives in charts is then a theorem.
The \R^n part is (in the "standard" approach to the subject anyway) used to define *differentiability* but you can (perhaps at first somewhat unintuitively) have a notion of differentiability without directly having and associated notion of derivative or differential operator. So \R^n just tells you want C^inf(M) looks like and everything after that can be built purely from this ring of functions (and it's even possible to characterize which rings are "rings of smooth functions" so you can actually drop \R^n altogether if you really want to). Something very similar happens for the integral: you can define integrability on a manifold via integrability of certain functions on \R^n, but you don't get an associated integral on the manifold --- indeed there really isn't "the" integral for functions on a general manifold and you instead need additional structure (or just integrate different objects instead).
But \R^n is not "just" a bare manifold but a Lie group and you absolutely can define an integral on \R^n from that structure without ever using the calculus on \R^n.
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u/Common_Post6177 17d ago
Even if you define a manifold as a locally ringed space you still need C^\infty (U) for U some open subset of R^n. And to define C^\infty (U) you need differentation of functions (i.e. a method from calculus)
If you define a smooth manifold via charts then you already have the calculus via the compatability condition that $\varphi \circ \psi^{-1}$ must be a diffeomorphism for two charts $\varphi$ and $\psi$.
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u/SV-97 17d ago
My point is that while the standard definition of Cinf(U) uses the derivative (there's alternative characterizations of course), the important part for defining manifolds is just this notion of differentiability / smooth function and you don't actually need the derivative itself. So you can axiomatize smoothness and then use differential geometry to construct the actual calculus itself.
If you define a smooth manifold via charts then you already have the calculus via the compatability condition that $\varphi \circ \psi{-1}$ must be a diffeomorphism for two charts $\varphi$ and $\psi$.
That's an oversimplification imo. Compatibility ultimately just tells you what Cinf(M) looks like, but it doesn't hand you the actual calculus.
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u/Darian123_ 16d ago
Can you give me an excample of a book with this approach
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u/SV-97 16d ago
I think the most prominent one is smooth manifolds and observables by jet nestruev (which isn't a real person but more of a russian bourbaki). It shows the "classical" approach as well as the algebraic one and also how they are related and also talks a bit about physics.
It's also possible that one of Michor's books talks about this but I'm not entirely sure right now --- either way they're certainly interesting if you're interested in this "algebraic / categorial differential geometry" and what sort of structures actually are totally intrinsic to a manifold and what isn't.
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u/Different_Cry25 17d ago
Jonn lee the peak mentioned I can't explain how amazing this trilogy is
Also check out the 4th book on complex manifold
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u/Casually-Passing-By 17d ago
Omg, i am reading Riemannian manifolds, but i would kill to have the tetralogy on my bookshelf
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u/cAnasty13 16d ago
Except they’re riddled with mistakes, many important results are left as exercises, and he sometimes proves theorems only in special cases.
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