ized ▁through ▁the ▁current ▁abstract ▁notion ▁of ▁a ▁manifold , ▁a ▁topological ▁space ▁where ▁the ▁smooth ▁structure ▁is ▁given ▁by ▁local ▁charts ▁on ▁the ▁manifold , ▁exactly ▁as ▁the ▁planet ▁Earth ▁is ▁mapped ▁by ▁at l ases ▁today . ▁Ch anges ▁of ▁coordinates ▁between ▁different ▁charts ▁of ▁the ▁same ▁region ▁are ▁required ▁to ▁be ▁smooth . ▁Just ▁as ▁cont our ▁lines ▁on ▁real - life ▁maps ▁encode ▁changes ▁in ▁elev ation , ▁taking ▁into ▁account ▁local ▁dist ort ions ▁of ▁the ▁Earth ' s ▁surface ▁to ▁calculate ▁true ▁distances , ▁so ▁the ▁Riemann ian ▁metric ▁describes ▁distances ▁and ▁areas ▁" in ▁the ▁small " ▁in ▁each ▁local ▁chart . ▁In ▁each ▁local ▁chart ▁a ▁Riemann ian ▁metric ▁is ▁given ▁by ▁smooth ly ▁assigning ▁a ▁ 2 × 2 ▁positive ▁definite ▁matrix ▁to ▁each ▁point ; ▁when ▁a ▁different ▁chart ▁is ▁taken , ▁the ▁matrix ▁is ▁transformed ▁according ▁to ▁the ▁Jacob ian ▁matrix ▁of ▁the ▁coordinate ▁change . ▁The ▁manifold ▁then ▁has ▁the ▁structure ▁of ▁a ▁ 2 - dimensional ▁Riemann ian ▁manifold . ▁ ▁Line ▁and ▁area ▁elements ▁T aking ▁a ▁local ▁chart , ▁for ▁example ▁by ▁project ing ▁onto ▁the ▁- plane ▁( ), ▁the ▁line ▁element ▁ ▁and ▁the ▁area ▁element ▁ ▁can ▁be ▁written ▁in ▁terms ▁of ▁local ▁coordinates ▁as ▁ ▁and ▁. ▁ ▁The ▁expression ▁ ▁is ▁called ▁the ▁first ▁fundamental ▁form . ▁ ▁The ▁matrix ▁ ▁is ▁required ▁to ▁be ▁positive - defin ite ▁and ▁to ▁depend ▁smooth ly ▁on ▁ ▁and ▁. ▁ ▁In ▁a ▁similar ▁way