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Manifold: Quiz

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Question 1: The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" ________ Rn.
Euclidean spaceVector spaceMathematicsDimension

Question 2: After a line, the ________ is the simplest example of a topological manifold.
PiCircleGeometryConic section

Question 3: Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are ________ functions.
Vector spaceComplex numberHolomorphic functionDerivative

Question 4: ________ further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables.
Johann Peter Gustav Lejeune DirichletFelix KleinCarl Friedrich GaussBernhard Riemann

Question 5: In the modern theory of manifolds, these notions correspond to ________ with constant negative and positive curvature, respectively.
Riemannian manifoldMetric tensorRiemannian geometryDifferentiable manifold

Question 6: In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the ________ of a specific dimension, called the dimension of the manifold.
Metric spaceEuclidean spaceCartesian coordinate systemVector space

Question 7: For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods ________ with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, and so differentiable on the manifold as a whole.
3-sphereFiber bundleGeneral linear groupDiffeomorphism

Question 8: On the other hand, a ________ can be sliced open by its 'parallel' and 'meridian' circles, creating a map with V = 1 vertex, E = 2 edges, and F = 1 face.
Stereographic projectionMöbius stripTorusManifold

Question 9: The study of manifolds combines many important areas of mathematics: it generalizes concepts such as ________ and surfaces as well as ideas from linear algebra and topology.
Differential calculusCurveElliptic curveAlgebraic geometry

Question 10: Informally, a manifold is a space that is "modeled on" ________.
MathematicsVector spaceEuclidean spaceDimension







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