where <tex2html_verbatim_mark>#math28#<tex2html_image_mark>#tex2html_wrap_inline1278#<SUB>(U, V)</SUB>(<tex2html_image_mark>#tex2html_wrap_inline1279#(<I>m</I>)×<tex2html_image_mark>#tex2html_wrap_inline1280#(<I>n</I>)) stands for the
tangent space to the manifold <tex2html_verbatim_mark>#math29#<tex2html_image_mark>#tex2html_wrap_inline1282#(<I>m</I>)×<tex2html_image_mark>#tex2html_wrap_inline1283#(<I>n</I>) at <tex2html_verbatim_mark>#math30#(<I>U</I>, <I>V</I>)∈<tex2html_image_mark>#tex2html_wrap_inline1285#(<I>m</I>)×<tex2html_image_mark>#tex2html_wrap_inline1286#(<I>n</I>) and so on. The projection of
therefore, is the product of the projection of the first component of
<tex2html_verbatim_mark>#math33#∇<I>F</I>(<I>U</I>, <I>V</I>) onto <tex2html_verbatim_mark>#math34#<tex2html_image_mark>#tex2html_wrap_inline1298#<SUB>U</SUB><tex2html_image_mark>#tex2html_wrap_inline1299#(<I>m</I>) and the projection of the
second component of <tex2html_verbatim_mark>#math35#∇<I>F</I>(<I>U</I>, <I>V</I>) onto <tex2html_verbatim_mark>#math36#<tex2html_image_mark>#tex2html_wrap_inline1304#<SUB>V</SUB><tex2html_image_mark>#tex2html_wrap_inline1305#(<I>n</I>).
In particular, we claim that the
projection <I>g</I>(<I>U</I>, <I>V</I>) of the gradient <tex2html_verbatim_mark>#math37#∇<I>F</I>(<I>U</I>, <I>V</I>) onto
<tex2html_verbatim_mark>#math38#<tex2html_image_mark>#tex2html_wrap_inline1311#<SUB>(U, V)</SUB>(<tex2html_image_mark>#tex2html_wrap_inline1312#(<I>m</I>)×<tex2html_image_mark>#tex2html_wrap_inline1313#(<I>n</I>)) is given by the pair of
defines a steepest descent flow on the manifold <tex2html_verbatim_mark>#math41#<tex2html_image_mark>#tex2html_wrap_inline1326#(<I>m</I>)×<tex2html_image_mark>#tex2html_wrap_inline1327#(<I>n</I>) for the objective function <I>F</I>(<I>U</I>, <I>V</I>).
