As the Earth became unable to sustain ten billion people, humanity took action. They built a giant space colony that orbits close to the Earth, and apparently took a Raygun Gothic design style along with them. There are five main points along this space station, each one hosting a group of countries, and an Orbital Union oversees it all.The year is Orbital Era 311. In the cluster of space colonies (or "Structure") called Kirkwood, located at the fifth Lagrangian point, 17-year old plucky Akiha Shishido is entering into an Arranged Marriage. She'd be the first one in her family to marry, but she's still in school and wants nothing to do with it. She and her little assistant, Imoko, get out of the way and end up walking into an abandoned space colony run by sentient AI Leopard, who's firmly on the side of crazy. At the same time, a mysterious girl named Honoka awakens and begins leading Akiha into more than she planned. Inter Colony Police member Itsuki Kannagi is also watching the events...Animated by Sunrise, the show is done by the same creative team behind Mai-HiME, Mai-Otome and Idolmaster: Xenoglossia.The title is a reference to the similarly named The Girl Who Leapt Through Time.
After the End: It is only vaguely alluded to throughout the series, but humanity nearly wiped itself out more than once, which is most likely why the Shishidou council decided to create the Brain Colonies in the first place. This works pretty well. If you like boxes.
A.I. Is a Crapshoot: Leopard is completely insane and, at his best, a major Jerkass. Nerval, on the other hand... is relatively sane, and not malevolent at all. He just takes "protect humanity" a bit too far.
Combining Mecha: Inverted: Bougainvillea and Mintao's joint QT Arms can split in two if needed.
Crazy-Prepared: The Department of Mysteries issues standard equipment to its officers "to act in every possible situation". This of course includes three differently-styled sets of female swimwear just in case the officer and two civilians come in a situation where an opportunity for Fanservice arises.
Dual Wielding: Takane goes for a rather realistic variety with a longsword in the main hand and a short sword/long dagger in the off-hand.
Dramatic Wind: Takane is so cool, she finds one on the Moon surface.
Egopolis: Leopard, being big enough to contain a city structure within him, takes pride in his city by, for example, naming all the animal species within him after himself. Crocodilius leopardus, anyone?
Panthera pardus leopardus?
Elseworld: Episode nine, out of nowhere, takes the main cast (including Leopard!) and sets them in a modern day world with a baseball team story. This episode serves as a sports anime-styled allegory explaining QT powers and their regulation, topics only hinted at in the rest of the series.
The Q-Tector bit may be slightly justified in that Leopard provided Akiha's. It doesn't explain why Itsuki's ICP-provided suit also experienced Clothing Damage, but... (actually, Itsuki herself mentions that her suit is affected because of the handcuffs - which sort of makes sense in a futuristic world)
Evil Costume Switch: Nami gets her own Q-Tector (in evil black and purple) when she goes to the dark side.
And then there's Nina, who looks and acts very much like an older, time-transplanted Sei Satou.
And the three female leads complete the same Sassy Action Girl heroine/Serious but fanservice-prone cool secret agent/short, mysterious weird girl triad as in Mai-HiME. This one is probably intentional.
Honoka's resemblance to Tsukasa probably means nothing.
Hikikomori: Leopard. He claims to be a "lone noble soul", but definitely shows some insecurity when he's accused of being a shut-in, not to mention his rather pained reaction to being exposed to the sun without any shade.
Also what the AIs are trying to turn all of humanity into. For our own safety, of course.
Nami is implied to be this, hiding away in her room. Though in her case it was a (mostly ignored) cry for help.
Indirect Kiss: Leopard gets a Luminescent Blush and says the trope name when Akiha drinks some of the tea he had been scanning. Yes he is a very large computer but as has been mentioned before he is quite insane.
Logic Bomb: Inverted: AI logic bombs a human. Leopard answers to the question "What are you?" with the counter question" Do you humans understand what you are?" and after a moment of awkward silence glees about how this question always short-circuits the human mind.
Akiha's "Imagine, imagine!" as she is about to place the Golden Orb into place may be a reference to Stellvia of the Universe, where it is Arisa's Catch Phrase, which she uses while handling heavy but delicate objects with mechanical arms. Also, the male student uniforms in Akiha's school are suspiciously similar to those on Stellvia.
Is it a coincidence that UL's cyberjack used for hacking into the base's mainframe looks exactly like R2-D2's?
At the beginning of episode five, Honoka checks Leopard's missing parts in a book labeled "Leopardnomicon".
The solar panels that are used to melt the ice off of Leopard during his battle with Nerval seem to suspiciously resemble a Solar System.
Is it just me, or does Nerval's "face open" form look a lot like Galvatron?
The line "What you can't do by yourself, you ask for help from others" was used in the finale of Code Geass.
Sora wo Kakeru Shoujo itself is referenced rather blatantly in Episode 0 of Yoku Wakaru Gendai Mahou, where the main character Koyomi (who has the same seiyuu as Imoko) is more or less forced onto a bus labeled Hako Bus, and is seen moments later wearing a blue version of the standard QT-Arms "Latex Space Suit".