By K. Kenmotsu
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Extra resources for Differential Geometry of Submanifolds
The fundamental objects are generalizations of vectors called tensors. Tensors can be viewed from many different perspectives. Mathematicians introduce tensors formally as a quotient of a certain module, while physicists introduce tensors using objects with many indices that transform in a specific way under a change of basis. Each definition is viewed disparagingly by the other camp, even though they are equivalent and both have their uses. We follow a middle approach here. 1 The tensor product To start, we define a new kind of vector product, called the tensor product, usually denoted by the symbol ⊗.
18) while the antisymmetric part Tasym of T is the antisymmetric tensor with components (Tasym )i j := 1 Ti j − T ji . 19) Evidently, for a (0, 2) tensor, T is the sum of its symmetric and antisymmetric parts. These ideas can be generalized to higher-order tensors, but the results are not as simple. Defining what is meant by “definite symmetry” for higher-order tensors requires some understanding of the representation theory of the symmetric group, which is beyond the scope of this work. Fortunately, for most purposes we need only consider the higher-order analogues of symmetric and antisymmetric tensors, and these are easy enough to describe using basic ideas of permutation theory, as follows.
Then we μ → λ ∧ μ. 57) for some linear functional f λ on n− p V . Given an inner product g, the Riesz lemma guarantees the existence of a unique (n − p)-vector λ such that g( λ, μ) = f λ (μ). 58) n− p The element λ ∈ V is called the Hodge dual or Hodge star of λ. 58) we may write λ ∧ μ = g( λ, μ)σ. 59) Written this way it is clear that the Hodge dual depends on both the inner product g and the choice of basis element σ for n V . At first sight the definition of the Hodge dual appears to be worthless, because it asserts the existence of an object but seems to give you no means of computing it.