if i check samples and it is hard to find out bacteria through microscope, could we use manganese content as geochemical evidence for bacteria in amount relatively
I would not favor this idea. Elevated Mn contents I either assume to be due to oxidizing conditions affecting the carbonate rocks resulting in the formation of "Fe-Mn limonite" or as an evidence for certain hydrothermal pulses. I found in many hydrothermal mineralizations abundant in calcareous gangue to have increased Mn contents in the mineral stages developing during the most recent part of the mineralizing process and most proximal to the surface.
frequently Mn and iron ore deposit are related to the part close to surface and we can look algae and lichens,...i confirm same idea of Pr. Dill an Pr. Kennth.
Manganese and iron enrichment can be a result of bacterial activity, so I would be skeptical about assuming that Mn is indicative of algae. Here's a reprint on bacterial Mn/Fe oxidation in a freshwater environment.