Abstract
In this paper we define nth order Hessian structures on manifolds and study them. In particular, whenn = 3, we make a detailed study and establish a one-to-one correspondence betweenthird-order Hessian structures and acertain class of connections on the second-order tangent bundle of a manifold. Further, we show that a connection on the tangent bundle of a manifold induces a connection on the second-order tangent bundle. Also we define second-order geodesics of special second-order connection which gives a geometric characterization of symmetric third-order Hessian structures.
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References
Abraham R and Marsden J E, Foundations of Mechanics, 2nd edition (Massachusetts: The Benjamin/Cummings Publishing Company Inc.) (1978)
Abraham R, Marsden J E and Ratiu T, Manifolds, tensor analysis and applications, first edition (Massachusetts: Addison-Wesley Publishing Company Inc.) (1983); and second edition (Springer-Verlag) (1987)
Ambrose W, Palais R S and Singer I M, Sprays (Anais da Academia Brasileira de Ciencias) (1960) vol. 32
Brickell F and Clark R S, Differentiable manifolds — an Introduction (London: Van Nostrand Reinhold Company) (1970)
David Kumar R, Second and higher order structures on manifolds, Ph.D. thesis (Hyderabad, India: University of Hyderabad) (1995) (unpublished)
David Kumar R and Viswanath K, Local expression of Hessian structures and dissections on manifolds,J. Ramanujan. Math. Soc. 2(2) (1987) 173–184
Fomin V E, Differential geometry of Banach manifolds, Russian version (Publishing House of Kazan University) (1983)
Flaschel P and Klingenberg W, Riemannsche Hilbert-mannigfalting keiten. Periodische Geodatische, LNM 282 (Berlin-Heidelberg, New York: Springer-Verlag) (1972)
Juhani Fiskaali, Sauli Luukkonen and Eljas Maatta, On the differential geometry of bounded projections on Banach spaces, Preprint (July, 1987)
Lang S, Differential manifolds (New York: Springer-Verlag) (1972)
Libermann P and Charles Michael Marle, Sympletic geometry and analytical mechanics (Dordrecht, Holland: D. Reidel Publishing Company) (1987)
Nelson E, Topics in dynamics I: Flows, Preliminary Informal Notes of University Courses and Seminars in Mathematics. Mathematical Notes (Princeton: Princeton University Press and the University of Tokyo Press) (1969)
Nelson Edward, Tensor analysis (Princeton, New Jersey: Princeton University Press and the University of Tokyo Press) (1967)
Nijenhuis A, Jacobi-type identities for bilinear differential concomitants of certain tensor fields I,Indag. Math. 17 (1955) 390–403
Nickerson H K, Spencer D C and Steenrod N E, Advanced Calculus (London: D. Van Nostrand Company Inc.) (1959)
Poor A Walter, Differential geometric structures (New York: Mc Graw-Hill Book Company) (1981)
Spivak Michael, A comprehensive introduction to differential geometry, second edition (Houston, Texas: Publish or Perish Inc.) (1970) vol.II
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David Kumar, R. Higher order Hessian structures on manifolds. Proc Math Sci 115, 259–277 (2005). https://doi.org/10.1007/BF02829657
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DOI: https://doi.org/10.1007/BF02829657