A local equivariant index theorem for sub-signature operators

In this paper, we prove a local equivariant index theorem for sub-signature operators which generalizes the Zhang's index theorem for sub-signature operators.


THE SUB-SIGNATURE OPERATORS
In this section, we give the standard setup (also see Section 1 in [24]). Let M be an oriented closed manifold of dimension n. Let E be an oriented sub-bundle of the tangent vector bundle TM. Let g TM be a metric on TM. Let g E be the induced metric on E. Let E ⊥ be the sub-bundle of TM orthogonal to E with respect to g TM . Let g E ⊥ be the metric on E ⊥ induced from g TM . Then (TM, g TM ) has the following orthogonal splittings Clearly, E ⊥ carries a canonically induced orientation. We identify the quotient bundle TM/E with E ⊥ .
Let Denote k = dimE and we assume k is even. Let {f 1 , · · · , f k } be an oriented (local) orthonormal basis of E. Set whereĉ(E, g E ) does not depend on the choice of the orthonormal basis. Let (2.10) Then we can check (2.12) where D * E is the formal adjoint operator of D E with respect to the inner product (2.3). Set From (2.11),D E is a formal self-adjoint first order elliptic differential operator on (M) interchanging ± (M, g E ).
Definition 2.1. The sub-signature operatorD E,+ with respect to (E, g TM ) is the restriction ofD E on + (M, g E ).
If we denote the restriction ofD E on ± (M, g E ) byD E,± , thenD * Recall that E is the subbundle of TM and that we have the orthogonal decomposition (2.1) of TM and the metric g TM . Let P E (resp. P E ⊥ ) be the orthogonal projection from TM to E(resp. E ⊥ ). Let ∇ TM be the Levi-Civita connection of g TM . We will use the same notation for its lift to (M). Set (2.14) Then ∇ E (resp.∇ E ⊥ ) is a Euclidean connection on E(resp.E ⊥ ), and we will use the same notation for its lifting on (E * )(resp. (E ⊥, * )). Let S be the tensor defined by Then S takes values in skew-adjoint endomorphisms of TM, and interchanges E and E ⊥ . Let {e 1 , · · · , e n } be an oriented (local) orthonormal base of TM. To specify the role of E, set {f 1 , · · · , f k } be an oriented (local) orthonormal basis of E. We will use the greek subscripts for the basis of E.
). (2.18) Let K be the scalar curvature of (M, g TM ). Let R TM (resp., R E , R E ⊥ ) be the curvature of ∇ TM (resp., ∇ E , ∇ E ⊥ ). Let {h 1 , · · · , h n−k } be an oriented (local) orthonormal base of E ⊥ . Now we can state the following Lichnerowicz type formula forD 2 E . From Theorem 1.1 in [24], we have Theorem 2.4. [24] The following identity holds,

A Local Even Dimensional Equivariant Index Theorem for Sub-Signature Operators
Let M be a closed oriented Riemannian manifold of even dimension n and φ an orientation-preserving isometry on M. Then the smooth map φ induces a mapφ = φ −1, * : ∧T * x M → ∧T * φ(x) M on the exterior algebra bundle ∧T * x M. LetD E be the sub-signature operator. We assume that dφ preserves E and E ⊥ and their orientations, thenφĉ(E, g E ) =ĉ(E, g E )φ. ThenφD E =D Eφ . We will compute the equivariant index We recall the Greiner's approach to the heat kernel asymptotics as in [19] and [4,5,13]. Define the operator given by maps u continuously to D (M × R, ∧T * M)) which is the dual space of c (M × R, ∧T * M)). We have Let (D 2 E + ∂ ∂t ) −1 be the Volterra inverse ofD 2 E + ∂ ∂t as in [5]. That is where R 1 , R 2 are smoothing operators. Let and k t (x, y) is the heat kernel of e −tD 2 E . We get Then Q 0 has the Volterra property, i.e., it has a distribution kernel of the form The parabolic homogeneity of the heat operatorD 2 E + ∂ ∂t , i.e. the homogeneity with respect to the dilations of R n × R 1 given by In the following, for g ∈ S(R n+1 ) and λ = 0, we let g λ be the tempered distribution defined by (3.8) Let C − denote the complex halfplane {Imτ < 0} with closure C − . Then: ) be a parabolic homogeneous symbol of degree m such that: (i) q extends to a continuous function on (R n × C − )\0 in such way to be holomorphic in the last variable when the latter is restricted to C − . Then there is a unique g ∈ S(R n+1 ) agreeing with q on R n+1 \0 so that: Let U be an open subset of R n . We define Volterra symbols and Volterra DOs on U × R n+1 \0 as follows.
] is a homogeneous Volterra symbol of degree l, i.e. q l is parabolic homogeneous of degree l and satisfies the property (i) in Lemma 2.3 with respect to the last n + 1 variables; (ii) The sign ∼ means that, for any integer N and any compact K, U, there is a constant C NKαβk > 0 such that for x ∈ K and for |ξ | + |τ | In what follows, if Q 0 is a Volterra DO, we let K Q 0 (x, y, t − s) denote its distribution kernel, so that the distribution K Q 0 (x, y, t) vanishes for t < 0.
) be a homogeneous Volterra symbol of order m and let g m ∈ C ∞ (U) ⊗ S (R n+1 ) denote its unique homogeneous extension given by Lemma 2.3. Then: Proposition 3.6. ( [5,13]) The following properties hold.
Parametrices. An operator Q is the order m Volterra DO with the paramatrix P then where R 1 , R 2 are smoothing operators.
We denote by M φ the fixed-point set of φ, and for a = 0, · · · , n, we let we may further assume that over the range of the domain of the local coordinates there is an orthonormal frame e 1 We shall refer to this type of coordinates as tubular coordinates.
where L 1 ∈ so(k) and L 2 ∈ so(n − k) The aim of this section is to prove the following result.

13)
where L 1 ∈ so(k), L 2 ∈ so(n − k) and Next we give a detailed proof of Theorem 3.9. Let Q = (D 2 E + ∂ t ) −1 . For x ∈ M φ and t > 0 set (3.14) Here we use a trivialization over ∧(T * M) about the tubular coordinates. Using the tubular coordinates, we have We mention the following result Similar to Theorem 1.2 in [15] and Section 2 (d) in [8], we have We will compute the local index in this trivialization. Let (V, q) be a finite dimensional real vector space equipped with a quadratic form.

Definition 3.11. The integer m is called as the Getzler order of Q. The symbol q (m) is the principal Getzler homogeneous symbol of Q. The operator Q
Let e 1 , . . . , e n be an oriented orthonormal basis of T x 0 M such that e 1 , · · · , e a span T x 0 M φ and e a+1 , · · · , e n span N φ x 0 . This provides us with normal coordinates (x 1 , · · · , x n ) → exp x 0 (x 1 e 1 +· · ·+x n e n ). Moreover using parallel translation enables us to construct a synchronous local oriented tangent frame e 1 (x), ..., e n (x) such that e 1 (x), · · · , e a (x) form an oriented frame of TM φ a and e a+1 (x), · · · , e n (x) form an (oriented) frame N τ (when both frames are restricted to M φ ). This gives rise to trivializations of the tangent and exterior algebra bundles. Write where A ij ∈ so(n).
Proof. In order to compute this differential form, we make use of the Chern root algorithm (see [22]). Assume that n = dimM and k = dimE are both even integers. As in [7], let L 1 ∈ so(k), L 2 ∈ so(n − k), we write (3.41) Then we obtain 1 4 1≤α,β≤k Then the left hand side of (3.40) is Now we consider the right hand side of (3.40), Then det 1 2 cosh Similarly, we have (3.47) On the other hand, Combining these equations, the proof of lemma 3.17 is complete.
To summarize, we have proved Theorem 3.9.

The Local Odd Dimensional Equivariant Index Theorem for Sub-Signature Operators
In this section, we give a proof of a local odd dimensional equivariant index theorem for sub-signature operators. Let M be an odd dimensional oriented closed Riemannian manifold. Using (2.19) in Section 2, we may define the sub-signature operatorsD E . Let γ be an orientation reversing involution isometric acting on M. Let dγ preserve E, E ⊥ and preserve the orientation of E, thenγτ (E, g E ) =τ (E, g E )γ , whereγ is the lift on the exterior algebra bundle ∧T * M of dγ . There exists a self-adjoint lift γ : Denotes byD + E the restriction ofD E to + (M, ∧(T * M)). We assume dimE = k is even, then (D E )ĉ(E, g E ) =ĉ(E, g E )(D E ) andĉ(E, g E ) is a linear map from kerD ± E to kerD ± E . Let Tr