Differential Geometry

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Arrow

01:44

@MartinSleziak I think it's nice to have separate rooms. Why do you want to create a single room?

@TedShifrin one of the applications if transversality is the existence of transverse pullbacks. However, there's a particular kind of pullback which I'm hoping always exists: given a smooth map $f:X\to Y$ and the tangent bundle $TY\to Y$, does the pullback of the latter along the former always exist?

Arrow

01:56

Ah, on second thought the tangent bundle is itself a submersion, so the pullback does indeed exist for any smooth map...

Arrow

02:13

Another question. I'm reading the [wiki entry on connections](https://en.wikipedia.org/wiki/Connection_(fibred_manifold)#Connection_as_a_horizontal_splitting) over surjective submersions. Doesn't a splitting of the short exact sequence displayed there amount to a choice of sections of the differentials at each point $p\in X$ which varies smoothly in $p$?

What's the geometric intuition relating parallel transport and sections of differentials?

It's like we're making a smooth choice of tangents upstairs which direct us "horizontally"...

1 hour later…

Martin Sleziak

03:29

@Arrow My suggestion was because this room (and to greater extent the other one mentioned there) has relatively low activity. In the transcript you can see that there were long periods with no messages.

Wider range of topics could mean more users, more activity. This would lead to more people who could notice the room and reply to the questions posted here.

But it was just a suggestion.

Periods of lower activity were noticed before by others, including the room creator:

May 25 '16 at 16:20, by Robert Cardona

This chat is quite dead. Perhaps we can start a small little reading group. Any recommendations on short interesting books?

But it is certainly true that the activity in this room increased recently, after Ted Shifrin started to visit it.

3 hours later…

Ted Shifrin

06:49

@Arrow: I don't know what you mean by "a choice of sections of the differentials..." To me connections are very geometric. For example, as you can see in my undergraduate differential geometry text (freely obtainable from my profile), the natural notion of connection for a surface in $\Bbb R^3$ is to differentiate the vector field as a vector-valued function and then project on the tangent space of the surface.

The splitting ideas for principal bundles are based on choosing a vector subspace (complementary to the vertical) consisting of tangent vectors coming from parallel transport in the more geometric setting I mentioned above. (A vector field is parallel along a curve if its covariant derivative is 0, i.e., if the tangential change along the curve is 0.)

But this is far more than can be discussed in a few comments in a chatroom.

3 hours later…

Arrow

09:38

@MartinSleziak maybe combining rooms is a good idea :)

2 hours later…

Balarka Sen

11:54

Let's merge them, @MartinSleziak.

It might turn out to be a fruitful idea after all.

Martin Sleziak

@BalarkaSen Ok, would somebody who is active in this area perhaps created the new room? Perhaps you?

Would it be for differential geometry, differential topology, algebraic topology, algebraic geometry?

Or is it suitable to add something else? Or omit some of the above?

Balarka Sen

I don't actually know how the merging procedure actually works.

I think "geometry & topology" suffices.

Martin Sleziak

BTW if it turns out that it does not work, we can unfreeze the old rooms.

@BalarkaSen I am not aware of any way of merging room in the sense that contents from various room would be combined together.

Balarka Sen

Oh.

Martin Sleziak

What I meant is creating a new room. And direct users of these rooms to the new one.

Balarka Sen

11:58

That sucks. There are useful contents in both of these rooms that I don't want to be demolished. I mean I can still link stuff from frozen rooms, but...

Martin Sleziak

I wonder whether general topology should stay separate. What would you think?

Balarka Sen

I wouldn't include it, no

Martin Sleziak

@BalarkaSen We might ask around, but I doubt there is a reasonable way to combine rooms. Does it changed that much?

Balarka Sen

What about this: we just change the original goal of the algebraic topology & homological algebra room to the broader goal of discussing geometry, topology and related stuff.

And redirect people from this room to there

I am just fond of that room, that's all :P

Martin Sleziak

I mean, after all algebraic topology was frozen not so long ago. And it would be frozen again if not kept alive artificially.

@BalarkaSen Personally, I am fine with such solution.

One additional advantage is that you - as a room owner - can change description and title of the room.

Balarka Sen

12:03

Right.

Martin Sleziak

Also if we compare room by activity, room info for algebraic topology shows 28k messages, while both algebraic geometry and differential geometry have around 1k.

Balarka Sen

Ya, it has far more activity than these rooms; one week it even exceeded the activity in the main chat.

I don't think including algebraic geometry is appropriate though.

Martin Sleziak

Would also commutative algebra be on topic, or is it more suitable for abstract algebra room? There used to by some commutative algebra chat rooms here and here

Balarka Sen

That's of a rather different flavor

So is com. alg.

Martin Sleziak

Ok, so suggestion is keep algebraic geometry separate and the additions to the algebraic topology room would be differential geometry (and perhaps differential topology - it does not have a separate room so far). Is that right?

I see:

in Geometry & Topology, 33 secs ago, by Balarka Sen

room topic changed to Geometry & Topology: Discussion in informal spirit [algebraic-topology] [differential-geometry] [differential-topology] [homological-algebra] [homotopy-theory]

Balarka Sen

12:08

Yep; I just changed the room name and the tags.

Martin Sleziak

Ok, so we will see what other users of this room say. But at the moment, the suggestion is to use that room for differential geometry. (And do not keep this one alive artificially.)

BTW if the numbers in the room info are correct, with this discussion we reached 1000 messages posted in this room.

Balarka Sen

Hah.

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