Explain the German invasion of France? How did the invasion affect France's alliance with Great Britain?

On September 3, 1939, in response to Hitler's invasion of Poland, Britain and France, both allies of the overrun nation declare war on Germany.

The first casualty of that declaration was not German-but the British ocean liner Athenia, which was sunk by a German U-30 submarine that had assumed the liner was armed and belligerent. There were more than 1,100 passengers on board, 112 of whom lost their lives. Of those, 28 were Americans, but President Roosevelt was unfazed by the tragedy, declaring that no one was to "thoughtlessly or falsely talk of America sending its armies to European fields." The United States would remain neutral.

As for Britain's response, it was initially no more than the dropping of anti-Nazi propaganda leaflets-13 tons of them-over Germany. They would begin bombing German ships on September 4, suffering significant losses. They were also working under orders not to harm German civilians. The German military, of course, had no such restrictions. France would begin an offensive against Germany's western border two weeks later. Their effort was weakened by a narrow 90-mile window leading to the German front, enclosed by the borders of Luxembourg and Belgium-both neutral countries. The Germans mined the passage, stalling the French offensive.