@TeresaLisbon Thank you very much

@Buraian @leslietownes Thank you 😊

I’ll try this then

@leslietownes Nice to see you here! Thank you for visiting the room, I hope we can see you here more often. Enjoying your contributions on the meta site as well. @Sarabsrimt It was good to be of help, once again.

@Buraian Now I remembered : I still have not edited that trigonometry answer you asked me to do so! I will do it now.

@TeresaLisbon I’m sure there are many more to come.

@Sarabsrimt Looking forward!

@TeresaLisbon Hi, I was wondering if you have got relatively free after Monday?

@Shashaank Yes, quite free after Monday. The work goes on, but the point is I'm better organized. So this time Monday, or 10:00 in the night would work on Monday, or for that matter any weekday.

@TeresaLisbon sorry did you mean that time would suit you for our discussion that I was saying? Are you free tomorrow?

Tomorrow will be possible in the night as well.

@TeresaLisbon ok let’s see tomorrow night then. You will not be free tomorrow morning, right?

@Shashaank Unlikely, Shashaank. I am scheduled to go somewhere tomorrow morning, and have some home tasks as well.

So the night should be good.

@TeresaLisbon Cool, that’s all right? By the way I was curious to know whether you did an MSc in maths or Btech from an IIT before joining for your phd

at ISI

And I had a very quick question in differential geometry? Can I ask?

In Manifolds particularly

@Shashaank My differential geometry has become very weak since my MSc. in Math at IISc, but if it has a little bit of PDE or analysis involved then I can help you. You can ask the question.

@TeresaLisbon I don’t think it requires any pde. Suppose you a 4d manifold. You can foliate it with hypersurfaces ( if you have read SR then the metric would be pseudo reimanian and you

could foliate it with 3D space like hypersurfaces). Now suppose you write the minkowski metric ( or any other Reimanian metric too), in Cartesian coordinates for a flat space ( its Reimann tensor will vanish). Now my question is whether you can change your coordinates such that a 3d submanifold is curved rather than bein curved ( so that Reimann tensor doesn’t vanish fir that submanifold)

That’s is the question is whether it’s possible that a submanifold of a flat space be curved

*rather than being flat

@Shashaank No, unfortunately I don't even know what a foliation is, so I can't be of help here.

But I think this question is worth a go on the main site!

That is, MSE.

Oops, I will try. It’s a long process to typeset using latex@TeresaLisbon

As long as you provide one or two links (for example, that the Riemann tensor vanishes, and about your background in Algebraic geometry and SR) then this actually makes for a nice question. In particular, it will be interesting for others to look at.

I don’t know actually if it will be nice or not. I thought it will be a trivial question so downvoated

My feeling is, that the last line "can a submanifold of a flat manifold be curved" is a real eye-catching line. It will attract attention, then again there could also be an affirmative answer which is not too difficult. I think it is worth a try, but you should mention the rough idea "can a submanifold of a flat manifold be curved". On your thought of it being trivial : Provide the context you need to, and if need be insert the links on the vanishing of the Riemannian metric, you'll be ok.

@TeresaLisbon btw foliate on just means that : a plane can be generated with a set of lines...we say the plane is foliated by lines.....

@TeresaLisbon I will try once I get a bit energy. Actually I am too lazy in writing Latex :)

Btw do you have any “ very easy “ text in pde or notes...I was thinking of reading a bit of pde...they are all over physics

@Shashaank Oh, that makes it clearer. Ok, now I see : I don't think a change of coordinates changes the curvature, I thought that the Riemann tensor is preserved under a change of coordinates? For the second part : for classical PDE consult the books of Evans, Strauss and John. For approaches via weak PDE see Kesavan's functional analysis. For a (large) standard reference for PDE see "Handbook of linear PDE for Engineers and Scientists" by Polyanin and Nazaikinskii.

None of them are "easy", but I would like to think that Evans provides more intuition than the others. John is more classical.

I could be wrong on the first part, of course. I still feel you should ask this on MSE, I'd like an answer to this one as well.

@TeresaLisbon Indeed Reimann tensor is invariant under change of coordinates but I am not asking whether the Reimann tensor of the whole manifold vanishes or not. It should vanish . If I begin with flat space do a coordinate transformation , the new metric should indeed give a vanishing Reimann tensor ( since the space was flat by construction)

What I am saying is that the Reimann tensor for the original metric is vanishing. Now I do a very precise coordinate transformation and ask whether the Reimann tensor of a submanifold of this space ( by putting one coordinate differential =0) an be non vanishing or not

@TeresaLisbon and thus the question- can a submanifold of a flat manifold be curved or not

@Shashaank Oh, I see : you're asking about a submanifold being curved! Then I can't say anything with too much confidence!

@TeresaLisbon exactly..maybe I will ask on mse

What is the relation between the Riemann tensor of a manifold and that of the submanifold? If you know something about this (it is restrictive, it is quite free etc.) that will give you an idea.

In fact, I don't see any relation. I'd like to think there is an example of your situation which exists.

Yeh I find pde tough ...odes

@TeresaLisbon @TeresaLisbon yes in physics there is paradox, Ehrenfrest paradox

@Shashaank I will look that up when I can. For now, I think you can try to post your question on MSE.

@TeresaLisbon sure I will try that

Till then see you tomorrow

Goodnight

See you and good night!