In the early morning of 10 April, leading elements of the Austrian army crossed the Inn River and invaded Bavaria. The early Austrian attack surprised the French; Napoleon himself was still in Paris when he heard about the invasion. He arrived at Donauwörth on the 17th to find the Grande Armée in a dangerous position, with its two wings separated by 75 miles (121 km) and joined together by a thin cordon of Bavarian troops. Charles pressed the left wing of the French army and hurled his men towards the III Corps of Marshal Davout. In response, Napoleon came up with a plan to cut off the Austrians in the celebrated Landshut Maneuver. He realigned the axis of his army and marched his soldiers towards the town of Eckmühl. The French scored a convincing win in the resulting Battle of Eckmühl, forcing Charles to withdraw his forces over the Danube and into Bohemia. On 13 May, Vienna fell for the second time in four years, although the war continued since most of the Austrian army had survived the initial engagements in Southern Germany.
Which river did the Austrians cross following their loss?
the Danube