This expository paper explores the torus through the lenses of topology, geometry, and dynamical systems. We begin by examining the fun-damental group of the torus, providing a foundation …

This paper offers full calculation of the torus’s shape operator, Riemann tensor, and related tensorial objects. In addition, we examine the torus’s geodesics by comparing a solution of the …

• In topology, a torus is any space homeomorphic to the torus. • This means there is a continuous bijective function to the torus, with a continuous inverse. • This is the sense in which ”donut” = …

Now consider the torus. Any point on the ”shell” of the torus can be identified by its position with respect to the center of the torus (the donut hole), and its location on the circular outer rim - …

We provide an example of a smooth one-parameter deformation of an Euclidean 2-torus, in fact an a ne bration, which corresponds to a continuous curve in the deformation space which is …

Theorem 5.1 Any connected abelian Lie group G is isomorphic to (R=Z)m ative integers k and m. In particular, if G is compact, it is isomorph up T is called a torus. By the previous theorem If T …

We develop a parametric representation of the torus by thinking of it as the surface swept out by a little circle rotating around on a big circle. Every point on the surface can be parameterized by …