Tangent Bundle Puns

Hi all. Looking for some resources for learning the topics in the title. I have done some reading in Nakahara and Frankel's books, but they still seem to complicated to me and I get lost. I have absolutely no intuition and its difficult to build the intuition in these books.

Anyone got some good basic resources for learning these and related topics (manifolds, Lie groups, bundles, etc)? Any advice for approaching these topics?

Thanks everyone!

Hey guys! So yeah, the title kinda says ir all, I’m trying to peove that for smooth manifolds M, N the tangent bundle T(M\times N) is diffeomorphic to TM \times TN. I don’t know how to describe the elements of the tangent bundles to make it convenient to see the diffeomorphism... Also, I supposed the tangent bundles have to be endowed with some sort of differentiable structure, how to do it? I don’t know if this helps bur I already proved that for all (p, q) in M\times N, the tangent space T_(p,q) (M\timesN) is isomorphic to the product of tangent spaces T_p M \times T_q N. Anyways, please help me out, any help is appreciated.

I think of tangent space at a point as a generalization of tangent lines/planes, or as locally linear approximations of the manifold at some point. Should I think of the tangent bundle as the collection of all those planes at every point of the manifold?

And how do I even begin to visualize the cotangent bundle? What exactly is cotangence? I usually think of the dual space of V as a space which is isomorphic to the original (in the finite dimensional cases at least), but has additional structure of multiplication with elements of V. I'm not exactly sure how to visualize the disjoint union of all the linear functionals at every point on the manifold.

Have a look at Smale's famous paper where he proved, among stronger statements, that a sphere eversion exists:

http://www.jstor.org/stable/1993205

In particular, he says at the very beginning that

> a homotopy of an immersion is called regular if at each stage it is regular and if the induced homotopy of the tangent bundle is continuous.

I understand the first requirement--that a regular homotopy is a one-parameter family of immersions--but I don't know what the motivation of the second requirement is. What happens if this requirement is dropped? What's the idea behind it? Any insight would be greatly appreciated!

[P.S., besides this I have overall found it to be a pretty neat paper of lesser than expected difficulty. The terminology with fiber bundles is a bit different from what I'm used to, but overall it's worth a read if you like topology.]

A (physical) field is a section of some fiber bundle E over your Lorentzian manifold M. You can consider E as a principle bundle with a Lie group G as it’s gauge group (or structure group). The field in a gauge theory is a connection on M, inducing a covariant derivative delta and further curvature R.

You can take the gauge group to be the multiplicative positive reals. In this case the curvature R is a real 2-form. Consider the decomposition M = T x N so that N is any Riemannian manifold. Then you can define E and B as 1-forms and 2-forms on N respectively satisfying the criteria that

R= B - dt ∧ E

Recall for an abelian gauge field you can compute the associated lagrangian using R and it’s hodge dual. In any case I skip some calculation steps but turns out this recovers the equations of classical electromagnetism. Notice how simple our assumptions are.

So if you ever forgot Maxwell’s equations you could derive them from first principles using only geometric assumptions!

EDIT 12/30: Ok, I will be choosing winners and PMing them with their keys! This may take me a bit. but I'll get it done asap!!!

UPDATE 2 1/02/2022: Still working on it! Doing this as fast as I can! Anything left over will go into another giveaway.

I have the following games, DLC, etc. leftover from various Humble Bundles and monthly Choices.

To enter: Please leave a comment listing 5 or more things you would like in order of preference, you will only win one item. I will award winners the top most choice I can. If all your choices are gone I will PM you to ask for your next preference.

I will not accept entries, trade requests, etc. via PM and anyone who PMs me with these will be disqualified!!!

The giveaway will close on Thursday, December 30, 2021 at 10pm EST.

I will be using THIS SITE to choose the winners. I was going to use Reddit Raffle, but they only allow 25 winners.

I will post the winner's name next to the item they received.

*** NOTES: There are 2 keys for Kingdom: Two Crowns and Generation Zero so there will be two winners for those games.

The following games are not Steam keys: Encodya (GOG) and Race the Sun (flippfly)

Google Docs link for list if you prefer.

​Part 1 Rewrite -- Original Part 1 --- Discord ---- YouTube --- My Patreon

Jake’s POV

“So, should we head down the mountain?” I asked.

“No, we think it would be best to just wait up here for the night, then head back down in the morning.” Suma explained.

“Well, we better get some firewood then, that way I don’t have to maintain this spell all night.” I suggested.

“Okay,” Vindicta said, “but.. what is firewood?” I sighed; it was going to be a long night atop this mountain.

-----------------------------------------------------------------------------------------------------------------------------------------

Suma’s POV

Turns out, firewood is exactly what it sounds like, wood specifically cut to be burned in a fire. “It feels like we are preparing for some kind of strange ritual.” Vindicta pointed out as she released her mana and let a bundle of log Jake had cut with magic fall to the ground. He had found a tree that “looked dry enough”, then used some kind of beam of magical energy to remove a branch and cut it into small sections.

“I know what you mean,” Von-Pac said as he carefully arranged the wood into the pattern Jake described beforehand. “I can’t say I have ever had to do anything like this before.”

“And he cast that magical spell without uttering a word.” Vindicta said. “You must have taught him well if he can use shadow casting, despite not having magic on his world.”

“I didn’t teach him a thing, in fact, I think he has probably taught me more about shadow casting than I ever could have taught him.” I said from my perch. My job was to be on lookout while everyone was working, just in case.

“What do you mean?” Vindicta asked.

“Jake taught himself shadow casting. In truth, I’m not sure I have ever heard him utter a single incantation.” I told them.

“I should be surprised at that, but I’m not.” Von-Pac said as he lit the small pile of wood aflame. Even from my perch I could feel the heat begin to increase. “Hot!” He exclaimed and fluttered back sharply, landing on another stack of wood behind him.

“How does he cast such powerful magic then?” Vindicta wondered.

“He has this way of imagining the spells and

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

This originally was a comment to one of u/happyegg1000 ‘s SLABS posts, but it actually ended up exceeding the permitted comment length (oops), so I’m just going to submit this as a post. This ties into the SLAB rabbit hole posts which I believe tie into the 2007/2008 global financial crisis, as well as the current financial crisis and the need for collateral, which appears to tie into GME, as stated in the original SLAB posts by happyegg etc.

Please note: As this was originally a comment, I wrote this up relatively quickly. It was meant as a means to put a lot of info together in one place in an efficient manner. I may have missed things, and this is where you all come in. Please go through the history of Navient/Sallie Mae and if I’ve missed anything of importance, feel free to add it in a comment, etc.

On to my original comment, which was posted in this thread https://reddit.com/r/Superstonk/comments/rpcyt6/the_slabs_rabbit_hole_part_3_revenge_of_the_slab/ and is presented below.

To u/happyegg1000 , other wrinkly brains, and even smooth brains- This makes me wonder what Navient will even do once they transfer out of the student loan market. IIRC their entire business is (was?) student loans.

Also, and perhaps more importantly, we need to note the history of Navient here.

so…here’s my quick gathering of info from Wikipedia. (I have omitted some things, not on purpose, but simply because this is a lot to go though. If you can, please help by going through their wiki pages and add any relevant info I may have missed.)

IMO, the first thing to know about Navient is that they are a relatively new name.

In fact, Navient used to be known as Sallie Mae and the Navient name was created in 2014:

> In 2014 Sallie Mae split into two entities: Navient (newly formed) and Sallie Mae.

(I remember because my private loans at the time were suddenly coming to me not from Sallie Mae anymore, and instead from Navient.)

Wiki page for reference:

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

And some of Sallie Mae’s history can be found here on wikipedia with cited sources:

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

Happy digging.

Edit - as you can see, some of this is/was rooted in the financial crisis of 2007/2008.

> For example, JC Flowers & Co’s unsuccessful bid to acquire Sallie Mae in 2007. (More on them below)

> Then…In 2008 Salle Mae completed a 31 billion dollar funding

Do your worst!

I'm surprised it hasn't decade.

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

Because she wanted to see the task manager.

Heard they've been doing some shady business.

but then I remembered it was ground this morning.

Edit: Thank you guys for the awards, they're much nicer than the cardboard sleeve I've been using and reassures me that my jokes aren't stale

Edit 2: I have already been made aware that Men In Black 3 has told a version of this joke before. If the joke is not new to you, please enjoy any of the single origin puns in the comments

BamBOO!

They’re on standbi

Chapter Ten

“Who the hell decided to let a human walk around the ship with a sidearm?” Said one of the marines to the left of the woman who had spoken initially.

“Holy fucking shit, he’s taller than any male I’ve ever met!” said the woman who had spoken initially.

“Ladies, I’m sorry to cut this short, but I’ve got to get back to the pod.” Interrupted Adam, trying to defuse the situation and squeeze by at the same time, only to have an armor clad arm block his escape through the doorway.

“I don’t think that’s gonna happen, cutie.” Said the marine in the lead menacingly, before continuing. “I think we’re gonna have to interrogate you, and make sure you’re not a threat to the ship and its crew. Earth isn’t yet pacified, ya know?”

"Come on Cadrilla, just let him leave, he's obviously allowed to be here." said the third marine, earning a withering glare from the first.

Adam stepped back from his blocked exit, letting out a deep sigh. If he had to pick a place to fight three on one, it would be either a bathroom or a laundry room. There was a lot of bones in the human- and Shil'- hand that you have to be careful not to break in a fight, and the surrounding area had lots of hard surfaces for throwing people into.

He mentally planned out his upcoming fight, knowing damn well he would lose to the three seven-foot-tall amazons, but not intending to go down without making them suffer first. He planned on striking the first one fast, catching her off guard. He reasoned he would try to disable her throat before delivering a couple strikes to her lower ribs before moving on to the second. For her, he planned on throwing his weight fully into the marine, attempting to throw her into her friend. For the third he would…

Then his train of thought was interrupted by a familiar voice from behind the women.

“Care to explain exactly why you bitches are harassing the newest member of my goddamn team?” Shouted Popper from the hallway. She stood at an imposing seven-foot-three, with her arms crossed over the chest of her suit, emblazoned with tattoo and sigil looking artwork that probably held some significance in Shil’vati culture. The marines turned to face her, and Adam took the opportunity for what it was, shoulder checking the lead marine on the way out.

“Hey what the fuck?" Shouted the lead marine incredulously.

"He just assaulted a member of the Shil’vati military, you’re just gonna let him get away with that?”

“Yes I fucking am, dick-for-brains!" Popper snarled

They were cooked in Greece.

A play on words.

Pilot on me!!

Christopher Walken

Nothing, he was gladiator.

Or would that be too forward thinking?

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "

I won't be doing that today!

Where ever you left it 🤷‍♀️🤭

This morning, my 4 year old daughter.

Daughter: I'm hungry

Me: nerves building, smile widening

Me: Hi hungry, I'm dad.

She had no idea what was going on but I finally did it.

Thank you all for listening.

[Removed]

There hasn't been a post all year!

You take away their little brooms

It was about a weak back.

When I got home, they were still there.