Someone please explain to me why Riven doesn't have a mana bar.
Riven is pretty easy to explain.
Auto-attacks are manaless.
Burst damage costs more and more mana the more front-loaded it makes your damage.
Veigar and LeBlanc have pretty obscene mana costs for their ability to front-load ~10 seconds of damage into 1 second.
Riven's combo is something like R + E + auto + Q + auto + W + auto + Q + auto + Q + auto + auto + R
cramming 10 seconds of burst into ~7 seconds.
Not that impressive compared to Jax (also a fighter/assassin) who is capable of landing a Leap Strike + Empowered Strike at the same time, fallowed emediatly by a sheen/tri-force AA with a Grandmaster's proc, both damaging instances a fraction of a second apart, in a window where you can activate Counterstike at the same time to unload all of his kit's damage in an instant.
Obviously Jax's combo does less damage than Riven's, but it's still cramming ~5 seconds of damage into 1 second.
Riven gets to be manaless because there's just a lot of time to react to her combo. You can move, cast your skills, unload your entire QWER rotation long before she can finish her's.
Jax just unleashes everything on you, skips away, then does it again 11 seconds later.
And again - to highlight the manaless nature of auto-attacks, both Jax's AA steroids don't cost mana, so given the oppertunity to dish out some sustained DPS over burst, he's not gated by mana when he finds that opportunity.
If you look at most alternate-resource champions they function in this way, especially if you compare them to other members of their archetype. Kennen and Vladimir are more on the sustained-damage, slow-combo direction over other mages, Garen's Judgement and Rumble's Flame-Spitter might as well be Tiamat auto-attacks, Aatrox, Yasou, and Tryndamere basically don't do anything but auto-attack. Akali is even pretty close to an auto-attacker with her set-up, requring on AAing to make use of her passive, splitting her Q into 2 parts, and requiring 3 ult casts (over just 1) to complete a full combo.
The other main consideration of mana costs are range. You can easily feel this with champions like Caitlyn, Ashe, Ezreal, Lux - if you want more range than their kit/AA provides, you spend some mana for a significant boost in range.
You'll generally notice a lot of manaless champions also have very minimal range. Most of them are melee, and in the case of Vlad and Kennen - ~650 range is not that much when you've got Lux, Ziggs, Xerath, and Morganna in the same archetype.
So again with Riven - she gets basically no range at all on anything but her ultimate's special, and she's basically an auto-attacker with an interesting take on an AA steroid. There's lots of ways to work around her, and a significant number of disadvantages her skills have.
Not exactly sure the reasoning for champions like Zed and Kat to be manaless.
Kat at least has a weak lanning phase reliant on actually using her manaless nature by poking things to break even, but haven't figured this one out.
TLDR: Riven is manaless because auto-attacks are manaless, and Riven basically auto-attacks things with her abilities.