Jacobi Fields in Geodesic Surface Congruences

Abstract

Using the Nambu-Goto string action in the space of the surfaces spanned by closed strings in a spacetime manifold, we investigated the geodesic surface equation in the space of surfaces joining two given strings and the geodesic surface deviation equation in geodesic surface congruence which yields a Jacobi field along a given geodesic surface, and singularities in geodesic surface congruences. In this paper, assuming that the singularity exists in geodesic surface congruences in a conformally symmetric manifold, we compute the Jacobi fields of the geodesic surface deviation equations and observe them.

Quantum Cohomologies of Symmetric Products

“…To be an element u in M(M, ; A) S k , there is an automorphism θ ∈ Aut(u) such that σ • u = u • θ for each element σ ∈ S k . Since the restriction π : → is a homeomorphism and the image u (z i ) ∈ , i = 1, 2, 3, there is a unique J X -holomorphic map u 1 : (Σ; z 1 , z 2 , z 3 ) → ( ; a 1 , a 2 , a 3 …”

Singularities and Thermodynamics of Geodesic Surface Congruences

“…In this section, we will pedagogically summarize the previous results Cho & Hong (2008;2010) on the expansion rate of stringy congruence in the early universe for the sake of completeness. We will consider the twist and shear motions in the next section.…”

“…The Large Hadron Collider at CERN is expected to be able to determine which cosmology among several scenarios including the stringy cosmology is viable through detecting the quark-gluon plasma state and its ensuing evolution of the tiny universe. This chapter is mainly based on the articles Cho & Hong (2007;2008) ;Cho (2010); Cho & Hong (2010).…”

“…The equation of state of the massive particle is different from that of the massless particle, and thus the massive particle phase is not the same as the massless particle one. Applying the string theory Green et al (1987); Polchinski (1999) to cosmology, we might investigate the expansion of the universe in terms of the Hawking-Penrose singularity in geodesic surface congruences for the time-like and null strings Cho & Hong (2008;2010). Taking an ansatz that the expansion of the stringy congruence is constant along the string coordinate direction, we derive the Raychaudhuri type equation, which is an evolution equation for the expansion, possessing correction terms associated with the stringy configurations.…”

Aspects of Today's Cosmology

“…The Large Hadron Collider at CERN is expected to be able to determine which cosmology among several scenarios including the stringy cosmology is viable through detecting the quark-gluon plasma state and its ensuing evolution of the tiny universe. This chapter is mainly based on the articles Cho & Hong (2007;; Cho (2010); Cho & Hong (2010).…”