We prove that the higher signature for any close oriented manifold is a homotopy invariant. 2000 MR Subject Classification 57N70, 57R95, 57R40

We define and study the signature and higher signatures of the quotient space of an S-action on a closed oriented manifold.

In this paper we consider pseudo projectively flat Riemannian manifold whose Ricci tensor S satisfies the condition S(X,Y ) = rT (X)T (Y ), where r is the scalar curvature, T is a non-zero 1-form defined by g(X, ξ) = T (X), ξ is a unit vector field and prove that the manifold is of pseudo quasi constant curvature, integral curves of the vector field ξ are geodesic and ξ is a proper concircular ...

In this report we introduce and motivate the problem of reconstructing shapes from partial information. An appropriate mathematical abstraction capturing the notion of a shape in three-dimensional space is a two-dimensional manifold. The concept of the topological type of a manifold plays an important role in reconstruction, and we present a synopsis of the pertinent definitions and results. We...

Integrating ideas from differential geometry and optimization, this chapter develops and applies novel parameterization-based framework to address optimization problems formulated on a smooth manifold, particularly those arise in computer vision. The framework views the manifold as a collection of local coordinate charts. It involves successive parameterizations of a manifold, carrying out opti...

We establish rigorous scaling laws for the average bursting time for bubbling bifurcations of an invariant manifold, assuming the dynamics within the manifold to be uniformly hyperbolic. This type of global bifurcation appears in nearly synchronized systems, and is conjectured to be typical among those breaking the invariance of an asymptotically stable hyperbolic invariant manifold. We conside...

We first present the natural definitions of horizontal differential, divergence (as an adjoint operator), and a $p$-harmonic form on Finsler manifold. Next, we prove Hodge-type theorem for manifold in sense that $p$-form is harmonic if only Laplacian vanishes. This viewpoint provides new appropriate definition vector fields geometry. approach leads to Bochner-Yano type classification based Ricc...

We construct Bott-type Floer homology groups for the sym-plectic manifold (T S 1 ; can) and Bott-type Morse homology groups for the energy functional on the loop space of S 1. Both objects turn out to be isomorpic to the singular homology of the loop space of S 1. So far our objects depend on all choices involved, but the above isomorphism suggests further investigation to show independence of ...

Algebraic substantivalism, as an interpretation of general relativity formulated in the Einstein algebra formalism, avoids the hole argument against manifold substantivalism. In this essay, I argue that this claim is well-founded. I first identify the hole argument as an argument against a specific form of semantic realism with respect to spacetime. I then consider algebraic substantivalism as ...

This note investigates the possibility of converses of the Weyl theorems that two conformally related metrics on a manifold have the same Weyl conformal tensor and that two projectively related connections on a manifold have the same Weyl projective tensor. It shows that, in all relevant cases, counterexamples to each of Weyl’s theorems exist except for his conformal theorem in the 4-dimensiona...