Showing papers on "Scalar curvature published in 2014"

Total Mean Curvature and Submanifolds of Finite Type

Abstract: Differentiable Manifolds Riemannian and Pseudo-Riemannian Manifolds Hodge Theory and Spectral Geometry Submanifolds Total Mean Curvature Submanifolds of Finite Type Biharmonic Submanifolds and Biharmonic Conjectures lambda-biharmonic and Null 2-type Submanifolds Applications of Finite Type Theory Additional Topics in Finite Type Theory

Optimization Techniques on Riemannian Manifolds.

Abstract: The techniques and analysis presented in this paper provide new methods to solve optimization problems posed on Riemannian manifolds. A new point of view is offered for the solution of constrained optimization problems. Some classical optimization techniques on Euclidean space are generalized to Riemannian manifolds. Several algorithms are presented and their convergence properties are analyzed employing the Riemannian structure of the manifold. Specifically, two apparently new algorithms, which can be thought of as Newton's method and the conjugate gradient method on Riemannian manifolds, are presented and shown to possess, respectively, quadratic and superlinear convergence. Examples of each method on certain Riemannian manifolds are given with the results of numerical experiments. Rayleigh's quotient defined on the sphere is one example. It is shown that Newton's method applied to this function converges cubically, and that the Rayleigh quotient iteration is an efficient approximation of Newton's method. The Riemannian version of the conjugate gradient method applied to this function gives a new algorithm for finding the eigenvectors corresponding to the extreme eigenvalues of a symmetric matrix. Another example arises from extremizing the function $\mathop{\rm tr} {\Theta}^{\scriptscriptstyle\rm T}Q{\Theta}N$ on the special orthogonal group. In a similar example, it is shown that Newton's method applied to the sum of the squares of the off-diagonal entries of a symmetric matrix converges cubically.

\( {E_d}_{(d)}\times {{\mathbb{R}}^{+}} \) generalised geometry, connections and M theory

Abstract: We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a d-dimensional manifold for all d ≤ 7. The theory is based on an extended tangent space which admits a natural \( {E_d}_{(d)}\times {{\mathbb{R}}^{+}} \) action. The bosonic degrees of freedom are unified as a “generalised metric”, as are the diffeomorphism and gauge symmetries, while the local O(d) symmetry is promoted to Hd, the maximally compact subgroup of Ed(d). We introduce the analogue of the Levi-Civita connection and the Ricci tensor and show that the bosonic action and equations of motion are simply given by the generalised Ricci scalar and the vanishing of the generalised Ricci tensor respectively. The formalism also gives a unified description of the bosonic NSNS and RR sectors of type II supergravity in d − 1 dimensions. Locally the formulation also describes M-theory variants of double field theory and we derive the corresponding section condition in general dimension. We comment on the relation to other approaches to M theory with Ed(d) symmetry, as well as the connections to flux compactifications and the embedding tensor formalism.

Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs

Abstract: In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvature-dimension inequalities on graphs, building upon previous work of several authors.

Nearly Starobinsky inflation from modified gravity

Abstract: We study inflation induced by (power-low) scalar curvature corrections to General Relativity. The class of inflationary scalar potentials $V(\ensuremath{\sigma})\ensuremath{\sim}\text{exp}[n\ensuremath{\sigma}]$, $n$ general parameter, is investigated in the Einstein frame, and the corresponding actions in the Jordan frame are derived. We found the conditions for which these potentials are able to reproduce viable inflation according to the last cosmological data and lead to large scalar curvature corrections that emerge only at a mass scale larger than the Planck mass. The cosmological constant may appear or be set equal to zero in the Jordan frame action without changing the behavior of the model during inflation. Moreover, polynomial corrections to General Relativity are analyzed in detail. When de Sitter space-time emerges as an exact solution of the models, it is necessary to use perturbative equations in the Jordan framework to study their dynamics during the inflation. In this case, we demonstrate that the Ricci scalar decreases after a correct amount of inflation, making the models consistent with the observable evolution of the Universe.

Generalized Curvature-Matter Couplings in Modified Gravity

Abstract: In this work, we review a plethora of modified theories of gravity with generalized curvature-matter couplings. The explicit nonminimal couplings, for instance, between an arbitrary function of the scalar curvature R and the Lagrangian density of matter, induces a non-vanishing covariant derivative of the energy-momentum tensor, implying non-geodesic motion and, consequently, leads to the appearance of an extra force. Applied to the cosmological context, these curvature-matter couplings lead to interesting phenomenology, where one can obtain a unified description of the cosmological epochs. We also consider the possibility that the behavior of the galactic flat rotation curves can be explained in the framework of the curvature-matter coupling models, where the extra terms in the gravitational field equations modify the equations of motion of test particles and induce a supplementary gravitational interaction. In addition to this, these models are extremely useful for describing dark energy-dark matter interactions and for explaining the late-time cosmic acceleration.

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Abstract: This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

The Volkov-Akulov-Starobinsky Supergravity

Abstract: We construct a supergravity model whose scalar degrees of freedom arise from a chiral superfield and are solely a scalaron and an axion that is very heavy during the inflationary phase. The model includes a second chiral superfield X, which is subject however to the constraint X 2 = 0 so that it describes only a Volkov–Akulov goldstino and an auxiliary field. We also construct the dual higher–derivative model, which rests on a chiral scalar curvature superfield R subject to the constraint R 2 = 0, where the goldstino dual arises from the gauge–invariant gravitino field strength as mn Dm n. The final bosonic action is an R+R 2 theory involving an axial vector Am that only propagates a physical pseudoscalar mode.

Modular curvature for noncommutative two-tori

Abstract: Starting from the description of the conformal geometry of noncommutative 2-tori in the framework of modular spectral triples, we explicitly compute the local curvature functionals determined by the value at zero of the zeta functions affiliated with these spectral triples. We give a closed formula for the Ray-Singer analytic torsion in terms of the Dirichlet quadratic form and the generating function for Bernoulli numbers applied to the modular operator. The gradient of the Ray-Singer analytic torsion is then expressed in terms of these functionals, and yields the analogue of scalar curvature. Computing this gradient in two ways elucidates the meaning of the complicated two variable functions occurring in the formula for the scalar curvature. Moreover, the corresponding evolution equation for the metric produces the appropriate analogue of Ricci curvature. We prove the analogue of the classical result which asserts that in every conformal class the maximum value of the determinant of the Laplacian on metrics of a fixed area is attained only at the constant curvature metric. Introduction In noncommutative geometry the paradigm of a geometric space is given in spectral terms, by a Hilbert space H in which both the algebra A of coordinates and the inverse line element D are represented, the latter being an unbounded self-adjoint operator which plays the role of the Dirac operator. The local geometric invariants such as the Riemannian curvature are obtained in the noncommutative case by considering heat kernel expansions of the form Tr(ae−tD 2 ) ∼ ∑ n≥0 an(a,D )t −d+n 2 where d is the dimension of the geometry. One may equivalently deal with the corresponding zeta functions. Thus, it is the high frequency behavior of the spectrum of D coupled with the action of the algebra A in H which detects the local curvature of the geometry. In this paper we shall analyze in depth a specific example, that of the noncommutative two torus Tθ whose differential geometry, as well as its pseudo-differential operator calculus, was first developed in [8] . To obtain a curved geometry from the flat one defined in [8], one introduces (cf. [7], [13]) a Weyl factor, or dilaton, which changes the metric by modifying the volume form while keeping the same conformal structure. Both notions of volume form and of conformal structure are well understood (cf. [9]), and we recall in §1 how one obtains the modified Dirac The work of the first named author was partially supported by the National Science Foundation award no. DMS-0652164. The work of the second named author was partially supported by the National Science Foundation awards no. DMS-0652167 and DMS-0969672. 1 ar X iv :1 11 0. 35 00 v1 [ m at h. Q A ] 1 6 O ct 2 01 1 2 CONNES AND MOSCOVICI operator for the new curved geometry. The starting point is the computation of the value at s = 0 of the zeta function Tr(a|D|−2s) for the 2-dimensional curved geometry associated to the dilaton h. Equivalently we are dealing with the term a2(a,D ) of the heat expansion. This computation was initiated in the late 1980’s (cf. [7]), and the explicit result which proves the analogue of the Gauss–Bonnet formula was published in [13]. It was then extended in [14] to cover the case of arbitrary values of the complex modulus τ (which was τ = i in [13]). In these two papers only the total integral of the curvature was needed, and this allowed one to make simplifications under the trace which are no longer possible in the case of arbitrary a, i.e. to compute the local expression a2(a,D ). The technical obstacles for the local computation were overcome by means of • the general rearrangement Lemma of §6.2 ; • the assistance of the computer. While the original computation of [7] was done by hand, the role of the computer assistance is to minimize the danger of a computational mistake in handling the large number of terms, about one thousand, which arise in the generalized pseudodifferential expressions involved in a2(a,D ). The complete calculation of a2(a,D ) was actually achieved in 2009 and announced (including by internet posting, although with some typos) at several conferences (Oberwolfach 2009 and Vanderbilt 2011). Moreover, the same computation was done independently in [15] and gave a confirmation of the result. The main additional input of the present paper consists in obtaining an explicit formula for the Ray-Singer log-determinant of D (which was left open in [7]). By calculating the gradient of the Ray-Singer log-determinant in two different ways, one obtains new geometric insight as well as a deep internal consistency relation between the different terms of the log-determinant formula. At the same time this elucidates the role of the intricate two operator-variable function occurring in its expression. We now briefly outline the contents of this paper, starting with the description of the local curvature functionals determined by the value at zero of the zeta functions affiliated with the modular spectral triples describing the curved geometry of noncommutative 2-tori. As in the case of the standard torus viewed as a complex curve, the total Laplacian associated to such a spectral triple splits into two components, one 4φ on functions and the other 4 φ on (0, 1)-forms, the two operators being isospectral outside zero. The corresponding curvature formulas involve second order (outer) derivatives of the Weyl factor, and as a new and crucial ingredient they involve the modular operator ∆ of the non-tracial weight φ(a) = φ0(ae −h) associated to the dilaton h. For 4φ the result is of the form a2(a,4φ) = − π 2τ2 φ0(a ( K0(∇)(4(h)) + 1 2 H0(∇1,∇2)( <(h) ) , (1) where ∇ = log ∆ is the inner derivation implemented by −h, 4(h) = δ 1(h) + 2<(τ)δ1δ2(h) + |τ |δ 2(h), < is the Dirichlet quadratic form <(`) := (δ1(`)) 2 + <(τ) (δ1(`)δ2(`) + δ2(`)δ1(`)) + |τ |(δ2(`)) , and ∇i, i = 1, 2, signifies that ∇ is acting on the ith factor. The operators K0(∇) and H0(∇1,∇2) are new ingredients, whose occurrence is a vivid manifestation of

Dark matter as a ghost free conformal extension of Einstein theory

Abstract: We discuss ghost free models of the recently suggested mimetic dark matter theory. This theory is shown to be a conformal extension of Einstein general relativity. Dark matter originates from gauging out its local Weyl invariance as an extra degree of freedom which describes a potential flow of the pressureless perfect fluid. For a positive energy density of this fluid the theory is free of ghost instabilities, which gives strong preference to stable configurations with a positive scalar curvature and trace of the matter stress tensor. Instabilities caused by caustics of the geodesic flow, inherent in this model, serve as a motivation for an alternative conformal extension of Einstein theory, based on the generalized Proca vector field. A potential part of this field modifies the inflationary stage in cosmology, whereas its rotational part at the post inflationary epoch might simulate rotating flows of dark matter.

Cosmological Attractor Models and Higher Curvature Supergravity

Abstract: We study cosmological α-attractors in superconformal/supergravity models, where α is related to the geometry of the moduli space. For α = 1 attractors (1) we present a generalization of the previously known manifestly superconformal higher curvature su- pergravity model (2). The relevant standard 2-derivative supergravity with a minimum of two chiral multiplets is shown to be dual to a 4-derivative higher curvature supergravity, where in general one of the chiral superfields is traded for a curvature superfield. There is a degenerate case when both matter superfields become non-dynamical and there is only a chiral curvature superfield, pure higher derivative supergravity. Generic α-models (3) inter- polate between the attractor point at α = 0 and generic chaotic inflation models at large α, in the limit when the inflaton moduli space becomes flat. They have higher derivative duals with the same number of matter fields as the original theory or less, but at least one matter multiplet remains. In the context of these models, the detection of primordial gravity waves will provide information on the curvature of the inflaton submanifold of the Kahler manifold, and we will learn if the inflaton is a fundamental matter multiplet, or can be replaced by a higher derivative curvature excitation.

Spherical and Hyperbolic Embeddings of Data

Abstract: Many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects. For many types of data, these dissimilarities are not euclidean (i.e., they do not represent the distances between points in a euclidean space), and therefore cannot be isometrically embedded in a euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-euclidean data onto surfaces of constant curvature. We aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical) or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space. This process is efficient enough to allow us to embed data sets of several thousand objects. We apply our method to a variety of data including time warping functions, shape similarities, graph similarity and gesture similarity data. In each case the embedding maintains the local structure of the data while placing the points in a metric space.

Non-branching geodesics and optimal maps in strong \(CD(K,\infty )\)-spaces

Abstract: We prove that in metric measure spaces where the entropy functional is $$K$$ -convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite second moments and this plan is given by a map. The results are applicable in metric measure spaces having Riemannian Ricci curvature bounded below, and in particular they hold also for Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below by some constant.

Comment on “ f ( R , T ) gravity”

Abstract: In Phys. Rev. D 84, 024020 (2011) Harko, Lobo, Nojiri and Odintsov presented a modified theory of gravitation, $f(R,T)$ gravity, where the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar and of the trace of the stress-energy tensor. In this Comment we correct the conservation equation of the stress-energy tensor, since we noticed that the authors missed an essential term which has consequences in the equation of motion of test particles. We thus correct the derivation of this equation of motion and comment on some of its consequences.

Correspondence of phase transition points and singularities of thermodynamic geometry of black holes

Abstract: We explore a formulation of the thermodynamic geometry of black holes and prove that the divergent points of the specific heat correspond exactly to the singularities of the thermodynamic curvature. We investigate this correspondence for different types of black holes. This formulation can also be applied to an arbitrary thermodynamic system.

Cosmological reconstruction and stability in $f(R,T)$ gravity

Abstract: We study the cosmological reconstruction of $$f(R,T)$$ gravity (where $$R$$ and $$T$$ denote the Ricci scalar and trace of the energy–momentum tensor) corresponding to the evolution background in FRW universe. It is shown that any cosmological evolution including $$\Lambda $$ cold dark matter, phantom or non-phantom eras and possible phase transition from decelerating to accelerating can be reproduced in this theory. We propose some specific forms of Lagrangian in the perspective of de Sitter and power law expansion history. Finally, we formulate the perturbed evolution equations and analyze the stability of some important solutions.

Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics

Abstract: We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogenuous Monge-Ampere equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.

An existence theorem for the Yamabe problem on manifolds with boundary

Abstract: Let (M,g) be a compact Riemannian manifold with boundary. We consider the problem (first studied by Escobar in 1992) of finding a conformal metric with constant scalar curvature in the interior and zero mean curvature on the boundary. Using a local test function construction, we are able to settle most cases left open by Escobar's work. Moreover, we reduce the remaining cases to the positive mass theorem.

Trace-anomaly driven inflation in modified gravity and the BICEP2 result

Abstract: We explore conformal-anomaly driven inflation in $F(R)$ gravity without invoking the scalar-tensor representation. We derive the stress-energy tensor of the quantum anomaly in the flat homogeneous and isotropic universe. We investigate a suitable toy model of exponential gravity plus the quantum contribution due to the conformal anomaly, which leads to the de Sitter solution. It is shown that in $F(R)$ gravity model, the curvature perturbations with its enough amplitude consistent with the observations are generated during inflation. We also evaluate the number of $e$-folds at the inflationary stage and the spectral index $n_\mathrm{s}$ of scalar modes of the curvature perturbations by analogy with scalar tensor theories, and compare them with the observational data. As a result, it is found that the Ricci scalar decreases during inflation and the standard evolution history of the universe is recovered at the small curvature regime. Furthermore, it is demonstrated that in our model, the tensor-to-scalar ratio of the curvature perturbations can be a finite value within the $68\%\,\mathrm{CL}$ error of the very recent result found by the BICEP2 experiment.

Intrinsic Polynomials for Regression on Riemannian Manifolds

Abstract: We develop a framework for polynomial regression on Riemannian manifolds. Unlike recently developed spline models on Riemannian manifolds, Riemannian polynomials offer the ability to model parametric polynomials of all integer orders, odd and even. An intrinsic adjoint method is employed to compute variations of the matching functional, and polynomial regression is accomplished using a gradient-based optimization scheme. We apply our polynomial regression framework in the context of shape analysis in Kendall shape space as well as in diffeomorphic landmark space. Our algorithm is shown to be particularly convenient in Riemannian manifolds with additional symmetry, such as Lie groups and homogeneous spaces with right or left invariant metrics. As a particularly important example, we also apply polynomial regression to time-series imaging data using a right invariant Sobolev metric on the diffeomorphism group. The results show that Riemannian polynomials provide a practical model for parametric curve regression, while offering increased flexibility over geodesics.

A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality

Abstract: Let $$\mathbb M $$ be a smooth connected manifold endowed with a smooth measure $$\mu $$ and a smooth locally subelliptic diffusion operator $$L$$ satisfying $$L1=0$$ , and which is symmetric with respect to $$\mu $$ . We show that if $$L$$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590 , then the following properties hold: The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.

Noether symmetry approach in Gauss–Bonnet cosmology

Abstract: We discuss the Noether Symmetry Approach in the framework of Gauss–Bonnet cosmology showing that the functional form of the F(R, 𝒢) function, where R is the Ricci scalar and 𝒢 is the Gauss–Bonnet topological invariant, can be determined by the presence of symmetries. Besides, the method allows to find out exact solutions due to the reduction of cosmological dynamical system and the presence of conserved quantities. Some specific cosmological models are worked out.

KERR–NEWMAN BLACK HOLES IN f(R) THEORIES

Abstract: In the context of f(R) modified gravity theories, we study the Kerr-Newman black hole solutions. We study nonzero constant scalar curvature solutions and discuss the metric tensor that satisfies the modified field equations. We conclude that, in the absence of a cosmological constant, the black holes (BHs) existence is determined by the sign of a parameter h dependent of the mass, the charge, the spin and the scalar curvature. Different values of this parameter lead to diverse astrophysical objects, such as extremal and marginal extremal BHs. Thermodynamics of BHs are then studied, as well as their local and global stability. We analyze these features in a large variety of f(R) models. We remark the main differences with respect to general relativity and show the rich thermodynamical phenomenology that characterizes this framework.

Phase transition and thermodynamical geometry for Schwarzschild AdS black hole in $AdS_5\times{S^5}$ spacetime

Abstract: We study thermodynamics and thermodynamic geometry of a five-dimensional Schwarzschild AdS black hole in $AdS_5\times{S^5}$ spacetime by treating the cosmological constant as the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential. It is found that the chemical potential is always negative in the stable branch of black hole thermodynamics and it has a chance to be positive, but appears in the unstable branch. We calculate scalar curvatures of the thermodynamical Weinhold metric, Ruppeiner metric and Quevedo metric, respectively and we find that the divergence of scalar curvature is related to the divergence of specific heat with fixed chemical potential in the Weinhold metric and Ruppeiner metric, while in the Quevedo metric the divergence of scalar curvature is related to the divergence of specific heat with fixed number of colors and the vanishing of the specific heat with fixed chemical potential.

Noether Symmetry Approach in Gauss-Bonnet Cosmology

Abstract: We discuss the Noether Symmetry Approach in the framework of Gauss-Bonnet cosmology showing that the functional form of the $F(R, {\cal G})$ function, where $R$ is the Ricci scalar and ${\cal G}$ is the Gauss-Bonnet topological invariant, can be determined by the presence of symmetries. Besides, the method allows to find out exact solutions due to the reduction of cosmological dynamical system and the presence of conserved quantities. Some specific cosmological models are worked out

Dirac and Plateau billiards in domains with corners

Abstract: Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C2-smooth Riemannian metrics g on a smooth manifold X, such that scalg(x) ≥ κ(x), is closed under C0-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.

Rho‐classes, index theory and Stolz’ positive scalar curvature sequence

Abstract: In this paper, we study the space of metrics of positive scalar curvature using methods from coarse geometry. Given a closed spin manifold M with fundamental group Γ, Stephan Stolz introduced the positive scalar curvature exact sequence. Higson and Roe introduced a K-theory exact sequence → K∗(BΓ) α − → K∗(C ∗ Γ) j − K∗+1(D ∗) → in coarse geometry. The K-theory groups in question are the home of interesting (secondary) invariants, in particular the rho-class ρΓ(g) ∈ K∗(D ∗) of a metric of positive scalar curvature. One of our main results is the construction of a map from the Stolz exact sequence to the Higson–Roe exact sequence (commuting with all arrows), using coarse index theory throughout. The main tool is an index theorem of Atiyah–Patodi–Singer (APS) type. Here, assume that Y is a compact spin manifold with boundary, with a Riemannian metric g which is of positive scalar curvature when restricted to the boundary (and π1(Y ) = Γ). One constructs an APSindex IndΓ(Y ) ∈ K∗(C ∗ Γ). This can be pushed forward to j∗(IndΓ(Y )) ∈ K∗(D ∗ Γ) (corresponding to the ‘delocalized part’ of the index). The delocalized APS-index theorem then states that j∗(IndΓ(Z)) = ρΓ(g∂Z) ∈ K∗(D ∗). As a companion to this, we prove a secondary partitioned manifold index theorem for ρ-classes.

Biharmonic maps into a Riemannian manifold of non-positive curvature

Abstract: We study biharmonic maps between Riemannian manifolds with finite energy and finite bi-energy. We show that if the domain is complete and the target of non-positive curvature, then such a map is harmonic. We then give applications to isometric immersions and horizontally conformal submersions.