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## Differential geometry and topology
In mathematics,
## Intrinsic vs. Extrinsic
Initially and up to the middle of the nineteenth century, differential geometry was studied from the The intrinsic point of view is more powerful, and for example necessary in relativity where space-time cannot naturally be taken as extrinsic. (In order then to define curvature, some structure such as a connection is necessary, so there is a price to pay.) The Nash embedding theorem shows that the points of view can be reconciled for Riemannian geometry, even for global properties. ## Technical requirements
The apparatus of differential geometry is that of
A differential manifold is a topological space with a collection of homeomorphisms from open sets to the open unit ball in
At every point of the manifold, there is the tangent space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point. For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of A vector field is a function from a manifold to the disjoint union of its tangent spaces, such that at each point, the value is a member of the tangent space at that point. A vector field is differentiable if for every differentiable function, applying the vector field to the function at each point yields a differentiable function. Vector fields can be thought of as time-independent differential equations. A differentiable function from the reals to the manifold is a curve on the manifold. This defines a function from the reals to the tangent spaces: the velocity of the curve at each point it passes through. A curve will be said to be a solution of the vector field if, at every point, the velocity of the curve is equal to the vector field at that point.
An alternating k-dimensional linear form is an element of the antisymmetric k'th tensor power of the dual V ## Riemannian geometryA special case of differential geometry is Riemannian manifolds (see also Riemannian geometry): geometrical objects such as surfaces which locally look like Euclidean space and therefore allow the definition of analytical concepts such as tangent vectors and tangent space, differentiability, and vector and tensor fields. The manifolds are equipped with a metric, which introduces geometry because it allows to measure distances and angles locally and define concepts such as geodesics, curvature and torsion. ## Symplectic topology
This is the study of See also the list of differential geometry topics. ## external linkshttp://xahlee.org/Periodic_dosage_dir/shape_space/curves_surfaces_palais.pdf A Modern Course on Curves and Surface, Richard S Palais, 2003 | |||||||

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