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A smooth embedding is an injective immersion f : M → N that is also a topological embedding, so that M is diffeomorphic to its image in N. An immersion is precisely a local embedding – that is, for any point x ∈ M there is a neighbourhood, U ⊆ M, of x such that f : U → N is an embedding, and conversely a local embedding is an ...
On-the-job training (widely known as OJT) is an important topic of human resource management. It helps develop the career of the individual and the prosperous growth of the organization. On-the-job training is a form of training provided at the workplace. During the training, employees are familiarized with the working environment they will ...
For example, the real projective space of dimension , where is a power of two, requires = for an embedding. However, this does not apply to immersions; for instance, R P 2 {\displaystyle \mathbb {R} \mathrm {P} ^{2}} can be immersed in R 3 {\displaystyle \mathbb {R} ^{3}} as is explicitly shown by Boy's surface —which has self-intersections.
As another example, there can be no local diffeomorphism from the 2-sphere to Euclidean 2-space, although they do indeed have the same local differentiable structure. This is because all local diffeomorphisms are continuous , the continuous image of a compact space is compact, and the 2-sphere is compact whereas Euclidean 2-space is not.
The notion of a closed immersion is local in the sense that f is a closed immersion if and only if for some (equivalently every) open covering = the induced map : is a closed immersion. [ 5 ] [ 6 ] If the composition Z → Y → X {\displaystyle Z\to Y\to X} is a closed immersion and Y → X {\displaystyle Y\to X} is separated , then Z → Y ...
Let M and N be differentiable manifolds and : be a differentiable map between them. The map f is a submersion at a point if its differential: is a surjective linear map. [1] In this case p is called a regular point of the map f, otherwise, p is a critical point.
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles.
Instructor-led training, [1] or ILT, is the practice of training and learning material between an instructor and learners, either individuals or groups. Instructors can also be referred to as a facilitator, who may be knowledgeable and experienced in the learning material, but can also be used more for their facilitation skills and ability to deliver material to learners.