During the mussaf on the high holy days we bow down to the floor. Do women also have this minhag. Did they also bow down in the temple when it was standing on hearing the kohen godol say the shem.
Rivivos Efraim 3:421 brings reasonings that woman should bow down on Yom Kippur and reasonings that woman should not bow down on Yom Kippur. He discusses whether woman bowed down in the temple and says that they did not as they were not there, however had a lady been in the courtyard of the temple and heard the Kohain Gadol say Hashem's name they would be required to bow down too. He concludes that everyone should do according to their Minhag.
Bowing on the High Holidays is a relatively modern practice that wasn't specifically endorsed or described by the Talmud or early halakhists. The idea is to mimic the Yom Kippur service of the Second Temple, when (m. Yoma 6) "the priests and the people who were standing in the courtyard, when they heard the explicit name, which came out of the mouth of the High Priest, would kneel and bow and fall upon their faces." One way of approaching this question, then, is by asking "would women have bowed in the Second Temple?" To which the answer is possibly no -- women didn't stand "in the courtyard," but instead in the "women's courtyard." However, David ben Solomon ibn Abi Zimra (T'shuvos RDBZ 810) says that everyone who could hear the name would bow, and those who couldn't hear the name, even if they were within the courtyard, would not bow. That would seem to imply, by removing the distinction of the courtyard's walls, that women who were close enough would also bow.
Joseph b. Soloveitchik gives significant evidence about early modern practice in Nefesh haRav (pg. 214): "Our teacher recalled that when he was in Lithuania he did not see women bowing on Yom Kippur, until he came to Berlin where he saw it for the first time, and then he began to think on why the women did not do so in Lithuania. It occurred to him that the Tosafists quote the Yerushalmi saying that only the 'near' would bow and not the 'far,' and he said that the 'near' are those within the courtyard and the 'far' are those without." However a note in the book says that Soloveitchik elsewhere endorsed the contradictory interpretation of RDBZ.
Analytically, I would connect this to women's practice in other prayer rituals. It's extremely rare for women's sections to uniformly avert/cover their eyes from Birkas Kohanim, for example. In many synagogues, women are used to standing and watching while men perform the rituals and rarely receive official informed guidance. I think Soloveitchik's argument is admirable -- he says elsewhere that RDBZ is correct but here he uses a contradictory logic, presumably to defend the practice of Lithuanian women. The female mimetic tradition is valuable and its oddities are worthy of limmud zechut. But I don't think he meant to criticize the women of Berlin, who did bow and whose tradition is just as valuable.