Empty DataFrame's rows do not have `name`

If I create a non-empty empty DataFrame, then I can access the index while using apply by accessing the name field.

import pandas as pd
pd.__version__  # '1.2.4'

df = pd.DataFrame({'val': [5,2,1]}, index=[1,2,3])
print(df.apply(lambda x: x.name, axis=1))

1    1
2    2
3    3
dtype: int64

However, this does not seem to be the case for an empty DataFrame.

df = pd.DataFrame(index=[1,2,3])
print(df.apply(lambda x: x.name, axis=1))

1   NaN
2   NaN
3   NaN
dtype: float64

Why is this the case and what can be done to achieve the non-empty behaviour?

Further context

In my case, I am actually interested in a behaviour similar to this:

def f(n):
    return [n, n+1]

df.apply(lambda x: f(x.name), axis=1, result_type='expand')

I would like to be able to join the result to the DataFrame afterwards.