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first of all consider the 2-dim. topological manifold case

My Question : is there any difference between

$$f \times g : R \times R \to R \times R$$

$$(x,y) \to (f(x),g(y))$$

and $$F : R^2 \to R^2$$

$$(x,y) \to (f(x,y),g(x,y))$$

Consider two topological manifolds the torus ##T^2## and the euclidean plane ##R^2## such that both of them for any point (p) that lies on each manifold can be represented by an ordered pair ##p=(a,b)##

Since ##R^2 = R \times R##

and ##T^2 = S^1 \times S^1##

and consider chart maps ##X(p)## for the two manifolds separately such that

##X : U \to D##

for ##M=R^2 , U \subset R^2 and D \subset R^2## same for ##T^2##

for the case of ##R^2##

$$M=R^2={ (a,b) \in R^2 }$$

choose ##U \subset R^2## and ##D \subset R^2##

and say that $$X : U \to D$$

$$(a,b) \to (f(a,b),g(a,b))$$

but for the torus ##T^2## we consider the notion of PRODUCT MANIFOLD

$$M=T^2={(a,b) \in T^2 : T^2 = S^1 \times S^1}$$

choose ##U \subset S^1 , D \subset R##

such that $$X : U_1 \to D$$

$$a \to X(a)$$ same for the other circle with ##Y## chart map

now consider the Cartesian product of the two chart maps $$X \times Y$$

$$X \times Y : U_1 \times U_2 \to D \times D$$

$$(a,b) \to (f(a),g(b))$$

for the manifold being the euclidean plane ##(R^2)##we used the chart map such that each coordinate is a function of the point which is function of two variables ##(a,b)## and for the manifold being the torus ##(T^2)## we used the chart map such that each coordinate is function of each point which is function of single variable .

The question is , can we use the method or the notion of PRODUCT MANIFOLD for the Euclidean Plane (Since ##R^2 = R \times R## ) same as it was used for the Torus ##T^2## ?

Example for some chart maps for both the manifolds

##M=R^2##

$$(a,b) \to (f(a,b),g(a,b))$$

$$(a,b) \to (\sqrt {a^2 + b^2} , \tan^{-1}(\frac a b))$$

for ##U=\{(a.b) \in R^2 : a \gt 0 \}## and ##D=\{(f(a,b),g(a,b)) \in R^2 :f(a,b) \gt 0 ,0 \gt g(a,b) \lt \pi \}##

and for ##M=T^2## if we consider the configuration space of the double pendulum being the Torus ##T^2## such that $$(a,b) \mapsto (f(a),g(b))$$

$$(a,b) \mapsto (\sin(a) , \sin(b))$$

for ##U=\{(a,b) \in T^2 : 0 \gt a \lt \frac{\pi }{2} , 0 \gt b \lt \frac{\pi }{2} \}##

Thanks.

<Moderation note: fixed some minor LaTex errors to improve readability>

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# I Chart coordinate maps of topological manifolds

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