  Double detonations in sub-Chandrasekhar mass carbon-oxygen white dwarfs with
helium shell are a potential explosion mechanism for a Type Ia supernova (SNe
Ia). It comprises a shell detonation and subsequent core detonation. The focus
of our study is on the effect of the progenitor metallicity on the
nucleosynthetic yields. For this, we compute and analyse a set of eleven
different models with varying core and shell masses at four different
metallicities each. This results in a total of 44 models at metallicities
between 0.01$Z_\odot$ and 3$Z_\odot$. Our models show a strong impact of the
metallicity in the high density regime. The presence of $^{22}$Ne causes a
neutron-excess which shifts the production from $^{56}$Ni to stable isotopes
such as $^{54}$Fe and $^{58}$Ni in the $\alpha$-rich freeze-out regime. The
isotopes of the metallicity implementation further serve as seed nuclei for
additional reactions in the shell detonation. Most significantly, the
production of $^{55}$Mn increases with metallicity confirming the results of
previous work. A comparison of elemental ratios relative to iron shows a
relatively good match to solar values for some models. Super-solar values are
reached for Mn at 3$Z_\odot$ and solar values in some models at $Z_\odot$. This
indicates that the required contribution of SNe Ia originating from
Chandrasekhar mass WDs can be lower than estimated in orevious work to reach
solar values of [Mn/Fe] at [Fe/H]$=0$. Our galactic chemical evolution models
suggest that SNe Ia from sub-Chandrasekhar mass white dwarfs, along with
core-collapse supernovae, could account for more than 80% of the solar Mn
abundance. Using metallicity-dependent SN Ia yields helps to reproduce the
upward trend of [Mn/Fe] as a function of metallicity for the solar
neighborhood. These chemical evolution predictions, however, depend on the
massive star yields adopted in the calculations.
