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In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways:
Therefore, the Euclidean space itself gives the first example of a manifold. The surface of a sphere such as the Earth provides a more complicated example. Note that the whole surface cannot be drawn on one map, but it can be covered by just a few maps, and hence the surface of the Earth is a (two-dimensional) manifold.
1 History
The first to have conceived clearly of curves and surfaces as spaces by themselves was possibly Carl Friedrich Gauss, the founder of intrinsic differential geometry with his theorema egregium. Bernhard Riemann was the first to do extensive work that really required a generalization of manifolds to higher dimensions. Abelian varieties were at that time already implicitly known, as complex manifolds. Lagrangian mechanics and Hamiltonian mechanics, when considered geometrically, were also naturally manifold theories, with a concept of generalized coordinates.
1.1 Intrinsic vs. extrinsic
The given characterisations are intrinsic to M: if we imagine a small insect on (or maybe better "in") M, with eyes that only see nearby points, we are describing it from the insect's point of view. It is also possible and very useful to describe a manifold from the point of view of an outside observer. For example if a fly is crawling on an orange, we can watch this from outside in three-dimensional space, while the fly is staying on the two-dimension surface of orange peel. This point of view is called extrinsic. It is historically prior to the intrinsic point of view. During the nineteenth century, first geometryGeometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible to proof, bu learned to consider that N dimensions were mathematically natural, with N > 3, and then that the intrinsic point of view was also geometrical. This was seen in a number of ways, for example when 'space' meant phase space in physicsPhysics (from the Greek, physikos , "natural", and physis , "Nature") is the science of Nature in the broadest sense. Physicists study the behavior and properties of matter in a wide variety of contexts, ranging from the sub-microscopic particles from whi, or 'geometry' meant curvatureCurvature is the amount by which an geometric object deviates from being flat''. The word flat might have very different meaning depending on the object considered (for curves it is a straight line and for surfaces it is a Euclidean plane). In this articl in Riemannian geometryIn mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics; i..
Therefore there are dual points of view to acquire on manifolds. They have a certain kind of intrinsic geometry, starting with their topologyTopology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study, talk. See also earth science, geography, human geography, g. They also have a geometry inside other spaces, an extrinsic geometry that depends on how they are 'mapped' into another space (think for example that every helixThis article is about the shape. See Helix (disambiguation) for other meanings. A helix (plural: helices is a twisted shape like a spring, screw or a spiral staircase. Helices are important in biology, as DNA is helical and many proteins have helical subs is the same line wrapped in different ways round cylinderThe word cylinder has several meanings. For the geometric object, see Cylinder (geometry . For the engine component, see Cylinder (engine . In firearms the cylinder is the rotating device that contains the firing chambers of a revolver. The phonograph cyls). Manifolds include familiar curves such as the circle, or surfaces in three-dimensional space that are locally smooth. They include many other possibilities that are harder to visualise, such as the Lie groups basic to mathematics and theoretical physics.
