Answer to Question #138194 in Differential Geometry | Topology for anjali g

Question

Let (a,b,c ) not equal to 0 be real constants. Show that the ellipsoid
E =((x,y,z) ∈R^3 : x^2/a^2 + y^2/b^2 + z^2/c^2 = 1) is a smooth surface.

Answer 1

We can take $a,b,c>0.$

We cover the ellipsoid by smooth charts of the form

$(x,y)\mapsto (x,y,c\sqrt{1-\frac{x^2}{a^2 }-\frac{y^2}{b^2 }})$ and $(x,y)\mapsto (x,y,-c\sqrt{1-\frac{x^2}{a^2 }-\frac{y^2}{b^2 }})$

where $\{(x,y)\in(-a,a)\times(-b,b)\}$

Similarly, $(y,z)\mapsto (a\sqrt{1-\frac{z^2}{c^2 }-\frac{y^2}{b^2 }},y,z)$ and $(y,z)\mapsto (-a\sqrt{1-\frac{z^2}{c^2 }-\frac{y^2}{b^2 }},y,z)$and lastly$(x,z)\mapsto (x,b\sqrt{1-\frac{z^2}{c^2 }-\frac{x^2}{a^2 }},z)$ and $(x,z)\mapsto (x,b\sqrt{1-\frac{z^2}{c^2 }-\frac{x^2}{a^2 }},z)$These charts are smooth clearly, also these maps takes into account of all possible values of the ellipsoid since on the ellipsoid at least one of $x,y \ or z \neq 0.$ Hence is covered in the first two $(z\neq 0)$ second two $(x\neq 0)$ or last two $(y\neq 0)$ .

Hence the ellipsoid can be covered by smooth atlas and hence is a smooth surface.