Lecture 7 47 Examples of symmetric spaces are given by Rn, Sn, and RP n , as well as by their direct products, about which we will say more momentarily. First, notice that the flat torus is symmetric, being the direct product of two symmetric spaces S1. However, the embedding of the torus into R3 produces a space which is not symmetric, since the isometry group does not act transitively on the points of the sur- face. In fact, the isometry group of the embedded torus of revolution (the bagel) in R3 is a finite extension of a one-dimensional group of rotations, while the isometry group of the flat torus is, as we saw last time, a finite extension of a two-dimensional group of translations. Hence the two surfaces are homeomorphic but not isometric. The flat torus R2/Z2 has no isometric embedding into R3, but it is isometric to the embedded torus in R4 given as the zero set of the two equations x1 2 + x2 2 = 1, x2 3 + x2 4 = 1. c. Remarks concerning direct products. Given any two sets X and Y , we can define their direct product, sometimes called the Carte- sian product, as the set of all ordered pairs (x, y): X × Y = { (x, y) | x X, y Y }. It is very often the case that if X and Y carry an extra structure, such as that of a group, a topological space, or a metric space, then this structure can be carried over to the direct product in a natural way. For example, the direct product of two groups is a group under pointwise multiplication, and the direct product of two topological spaces is a topological space in the product topology. If X and Y carry metrics dX and dY , then we can put a metric on X × Y in the same manner as we put a metric on R2, by defining d((x, y), (x , y )) = dX(x, x )2 + dY (y, y )2. If there are geodesics on X and Y , we can define geodesics on X × Y , and hence can define the geodesic flip, which can be shown to satisfy the formula I(x,y)(x , y ) = (Ix(x ), Iy(y )).
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