Short answer: absolutely not.
This is a fairly contentious issue, but people have allowed their national biases to cloud their judgement. There are generally two things you will hear on the matter: either that the AK is a shitty clone of the StG, or that Mikhail Kalashnikov just happened to come up with the same idea as the Germans at around the same time (the official, and obviously a little embellished, Soviet perspective). Both are incorrect.
Kalashnikov began developing what would become the AK-47 during WWII, after being wounded in battle. He was a tank crewman, and wanted to design a weapon that could be used by tank crews and infantry alike - a truly universal one. Initial designs were for a submachine gun, but it wasn't until the MKb 42 (H) and StG 44 were encountered that the AK took its form as a rifle - almost certainly, he was inspired by its success. This is about as far as the 'copy' theory holds any truth whatsoever. Let's look at why...
There is no denial that the two rifles look very similar in profile. They share the same general layout - 30-round magazine ahead of the trigger guard, pistol grip configuration, short sighting radius with raised front and rear sights, gas system tapped from the top of the barrel - but this is about where the similarities end. The AK's mag release is a typical Russian paddle between the mag and trigger guard, versus the left-side push-button on the StG (an upscaled carry-over from the MP 40). The AK's fire selector and safety are the same thing, and take the form of a large dustcover on the right side which serves as a three-position switch. The StG has a push-button fire selector above the trigger and a switch-type safety above the grip - two separate controls, compared to the AK's one.
The StG's dust cover is the sprung door type later used on most LMGs and the M16 family of rifles, serving no other function than to simply keep dust out of the weapon. The StG's cocking handle is on the left hand side, the AK's on the right. Both reciprocate. The StG is constructed mostly of stamped steel; early AK-47s are milled, with stamped variants only coming into major use later (and of much simplified construction compared to the StG, although it is worth noting that the Soviets did employ some of the German weapons manufacturers involved in the MP 40 and StG 44 for their knowledge of working with stamped steel for firearms construction).
The AK disassembles by pressing a thumblatch attached to the recoil
spring guide, which releases the top cover. The gas block is removed by
means of a simple latch. The StG's stock is removed by driving out a pin, and it comes free along with the recoil spring. The receiver splits into upper and lower parts on a hinge - similarly to the
PPSh-41 and most Western rifles since, including the FN FAL and M16 - and the cocking handle must be removed in order to remove the bolt carrier group. Clearly, the two rifles disassemble in a completely different manner.
Both rifles fire from a closed bolt, and rely on gas tapped from the barrel to cycle the action. The Soviets had operational piston-type gas systems before the Germans brought them into service - in fact the Germans copied the entire SVT-40 gas system to create the G/K 43 as the annular gas trap (or "bang" system) on the G 41 had proven extremely unsatisfactory, becoming easily clogged, easily damaged, and prone to short strokes (it was also mechanically overcomplicated and costly to produce).
Both rifles are select-fire, but the way in which their FCGs operate is totally different. In fact, the AK has more in common with two American rifles than it does the StG; Kalashnikov was heavily inspired by the Remington Model 8, an early self-loading rifle which featured the same dust cover safey and general receiver layout as the AK. The FCG was based on that of the M1 Garand.
The AK has a rotating bolt, the StG a tilting bolt - the later FN FAL shared this, along with many other features. The SVT-40 also had a titlting bolt, so you could actually argue that its gas system wasn't the only thing the Germans were impressed by.
In short - the operating mechanisms of these two rifles are about as separated as it is possible for two weapons of the same class to be.
Curved magazines are a simple way to increase capacity while decreasing size, and the arrival by almost all parties at the conclusion that 30 rounds is the ideal magazine size is no coincidence. It is a nice, round number which balances the need for firepower with the need for portability.
Both rifles use pretty standard sights for their time, there is no question of one copying the other as this was simply the go-to sight layout in the late 1940s (unless you hailed from America, where aperture-based sights have always been more popular).
So really, if you give it some thought, implying the AK was copied from the StG is like saying the M16 was copied from the FAL. It's ridiculous. However, the AK was almost certainly inspired by the StG... as many other weapons have been inspired by the AK. That's how firearms design works, folks; you take something that's good, and you see if you can make it better.