A Brief Introduction to Index Functions

Chungen Liu
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Chungen Liu: Guangzhou University, School of Mathematical and Information Science

Chapter Chapter 2 in Index theory in nonlinear analysis, 2019, pp 23-34 from Springer

Abstract: Abstract For n ∈ ℕ $$n\in \mathbb {N}$$ , we recall that the symplectic group is defined as Sp ( 2 n ) ≡ S p ( 2 n , ℝ ) = { M ∈ ℒ ( ℝ 2 n ) ∣ M T J M = J } , $$\displaystyle \mathrm {Sp}(2n)\equiv Sp(2n,\mathbb {R})= \{M \in \mathcal L(\mathbb {R}^{2n}) \mid M^{T}JM=J \}, $$ where 0 − I n I n 0 $$\left ( \begin {array}{cc} 0 \ \ & -I_{n}\\ I_{n} & 0 \end {array} \right )$$ , I n is the identity matrix on ℝ n $${\mathbb R}^{n}$$ , and ℒ ( ℝ 2 n ) $$\mathcal L({\mathbb R}^{2n})$$ is the space of 2n × 2n real matrices. Without confusion, we shall omit the subindex of the identity matrices. For τ > 0, suppose H ∈ C 2 ( S τ × ℝ 2 n , ℝ ) $$H \in C^{2}(S_{\tau } \times \mathbb {R}^{2n}, \mathbb {R})$$ , where S τ ≡ ℝ ∕ ( τ ℤ ) $$S_{\tau } \equiv \mathbb {R}/(\tau \mathbb {Z})$$ .

Date: 2019
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DOI: 10.1007/978-981-13-7287-2_2

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