Identify the compound

Decacarbonyldimanganese $(0)$ is: $\mathrm{Mn_2(CO)_{10}}$

It consists of two manganese atoms bonded together, with a total of 10 carbonyl ligands.

Determine the environment around each Mn atom

In $\mathrm{Mn_2(CO)_{10}}$, the structure is:

• Each Mn is bonded to 5 CO ligands
• The two Mn atoms are also bonded to each other by an Mn--Mn bond

So, around each manganese, the coordination count is: $5\text{ CO ligands} + 1\text{ Mn atom} = 6$

Thus, each Mn has six neighboring atoms/groups around it.

Infer the geometry

A coordination number of 6 generally corresponds to octahedral geometry.

Here, each Mn is surrounded octahedrally by:

• five carbonyl ligands
• one manganese atom

Check the options

• A: Trigonal bipyramidal $\rightarrow$ coordination number 5, so incorrect
• B: Square pyramidal $\rightarrow$ coordination number 5, so incorrect
• C: Square planar $\rightarrow$ coordination number 4, so incorrect
• D: Octahedral $\rightarrow$ coordination number 6, correct

Final answer

The coordination geometry around each manganese in $\mathrm{Mn_2(CO)_{10}}$ is: $\boxed{\text{Octahedral}}$