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If (M, g) is the unit sphere S n with its usual round metric and (N, h) is the sphere of radius 2 with its usual round metric, both thought of as subsets of the ambient coordinate space R n+1, then the "expansion" map φ : R n+1 → R n+1 given by φ(x) = 2x induces a geodesic map of M onto N.
Georgia is a state in the Southeastern United States in North America.The Golden Isles of Georgia lie off the coast of the state. The main geographical features include mountains such as the Ridge-and-valley Appalachians in the northwest, the Blue Ridge Mountains in the northeast, the Piedmont plateau in the central portion of the state and Coastal Plain in the south.
The stereographic projection maps the -sphere onto -space with a single adjoined point at infinity; under the metric thereby defined, {} is a model for the -sphere. In the more general setting of topology , any topological space that is homeomorphic to the unit n {\displaystyle n} -sphere is called an n ...
The coastline of Georgia about 315 kilometres (196 mi): [14] of the coastline, 57 kilometres (35 mi) is the coastline of Ajaria (Ajara) [15] and 200 kilometres (120 mi) is the coastline of Abkhazia. [16] Georgia has an Exclusive Economic Zone of 21,946 km 2 (8,473 sq mi) in the Black Sea.
Conformal symmetries of a sphere are generated by the inversion in all of its hyperspheres. On the other hand, Riemannian isometries of a sphere are generated by inversions in geodesic hyperspheres (see the Cartan–Dieudonné theorem.) The Euclidean sphere can be mapped to the conformal sphere in a canonical manner, but not vice versa.
Georgia borders the North Atlantic Ocean in the Southeastern United States. Georgia was the fourth of the original 13 states to approve the Constitution of the United States of America on January 2, 1788. Georgia joined the Confederate States of America during the American Civil War from 1861 to 1865, and was readmitted to the Union in 1870.
Georgia [c] is a country in Eastern Europe and West Asia. [14] [15] [16] It is part of the Caucasus region, bounded by the Black Sea to the west, Russia to the north and northeast, Turkey to the southwest, Armenia to the south, and Azerbaijan to the southeast. Georgia covers an area of 69,700 square kilometres (26,900 sq mi). [17]
Cartography has a long tradition in Georgia. Scientific and cartographic depictions of Georgia by Georgian scholars date back to the first half of the 18th century. Prince Vakhushti of Kartli, a family member of King Vakhtang VI of Kartli (the core region of Georgia), prepared a geographical description of Georgia and adjacent territories as well as geographical atlases.