Step 1: What is asked.
We need the river under which India's first underwater metro rail tunnel was built.
Step 2: Why it is special.
Building a tunnel under a river is a tough engineering task, since it must handle high water pressure and mud.
Step 3: The project.
This underwater metro is part of the Kolkata Metro, made to link the two sides of the river that splits the city.
Step 4: The river.
The tunnel was built beneath the Hooghly River, making it the first of its kind in India.
Step 5: Remove others.
Yamuna, Ganga, and Gomti are not the river for this tunnel.
Step 6: Conclusion.
So the river is the Hooghly.
Answer: Hooghly