Concept of Manifold;

Defination of manifold:

The concept of a manifold is in essence a generalization of the idea, first formulated in mathematical terms by Gauss, underlying the usual procedure  used in cartography (i.e. the drawing of maps of the earth's surface, or  portions of it). The reader is no doubt familiar with the normal cartographical process:  

               The region of the earth's surface of interest is subdivided into (possibly overlapping) subregions, and the group of people whose task it is to draw the  map of the region is subdivided into as many smaller groups in such a way  that:  

(i) each subgroup of cartographers has assigned to it a        particular subregion (both labelled i, say); and  

(ii) if the subregions assigned to two different groups (labelled i and j say) intersect, then these groups must indicate accurately on their maps the  rule for translating from one map to the other in the common region (i.e.  

region of intersection). (In practice this is usually achieved by giving beforehand specific names to sufficiently many particular points (i.e.  land-marks) of the original region, so that it is immediately clear which  points on different maps represent the same point of the actual region.)