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In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there.
In particular, a C k-atlas that is C 0-compatible with a C 0-atlas that defines a topological manifold is said to determine a C k differential structure on the topological manifold. The C k equivalence classes of such atlases are the distinct C k differential structures of the manifold. Each distinct differential structure is determined by a ...
A topological manifold that is in the image of is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold". Thus given two categories, the two natural questions are:
The exterior derivative of a totally antisymmetric type (0, s) tensor field with components A α 1 ⋅⋅⋅α s (also called a differential form) is a derivative that is covariant under basis transformations. It does not depend on either a metric tensor or a connection: it requires only the structure of a differentiable manifold.
Nevertheless, these solutions still have enough structure that they may be completely described. The first observation is that, even if f 1 and f 2 are two different solutions, the level surfaces of f 1 and f 2 must overlap. In fact, the level surfaces for this system are all planes in R 3 of the form x − y + z = C, for C a constant. The ...
If a function is differentiable at a point, its differential is given in coordinates by the Jacobian matrix. However, a function does not need to be differentiable for its Jacobian matrix to be defined, since only its first-order partial derivatives are required to exist.
In differential geometry, a G-structure on an n-manifold M, for a given structure group [1] G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G -structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields .
The differential ′ is a linear operator that maps an n × n matrix to a real number. Proof. Using the definition of a directional derivative together with one of its basic properties for differentiable functions, we have