Step 1: Start with the easiest statement to verify, Statement III.
Deuterium oxide, $\text{D}_2\text{O}$, is the scientific name for heavy water, used in nuclear reactors, and has nothing to do with hardness of water, so Statement III is false.
Step 2: Check Statement II.
Hard soaps are actually sodium or potassium salts of long chain fatty acids like stearic or palmitic acid, not sulphonic acids. Sulphonic acid salts describe synthetic detergents instead, so Statement II is also false.
Step 3: Check Statement I with the chemistry of hard water.
Hard water carries dissolved calcium and magnesium ions, and soap reacts with them to form an insoluble scum: \[ 2\text{C}_{17}\text{H}_{35}\text{COONa} + \text{Ca}^{2+} \to (\text{C}_{17}\text{H}_{35}\text{COO})_2\text{Ca}\downarrow + 2\text{Na}^+ \] This scum formation uses up soap before it can produce lather, so soap genuinely struggles to foam in hard water, making Statement I true.
Step 4: Combine the results.
With only Statement I holding up under scrutiny, that is the sole correct statement. \[ \boxed{\text{Statement I only}} \]