Phương trình đường trắc địa cực tiểu trên đa tạp với mật độ các đường trắc địa trên mặt phẳng với mật độ tuyến tính
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Abstract
The article generalizes the notion of geodesic curves in manifolds with density and presents their equations based on Christoffel symbols. Accordingly, we prove that on plane with linear density, of all curves joining the two points of p and q, the curve minimizes density arc-length if and only if its curvature equals 0.
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References
[1]. M. P. D. Carmo (1976), Differential geometry of curves and surfaces, Prentice-Hall, Engle-wood Cliffs, NJ.
[2]. R. Corwin, N. Hoffman, S. Hurder, V. Sesum, and Y. Xu (2006), “Differential geometry of manifolds with density”, Rose-Hulman Und. Math. J., 7 (1).
[3]. A. Cannas Da Silva (2001), Lectures on Symplectic Geometry, Spriger – Verlag – New York – Berlin –- Heidelberg – Tokyo.
[4]. M. Gromov (2003), “Isoperimetry of waists and concentration of maps”, Geom. Funct. Anal., No. 13, P. 178-215.
[5]. C. Ivan, H. Stephanie, A. Vojislav, And Y. Xu (2004), “Double bubbles in Gauss space and highdimensional spheres, and differential geometry of manifolds with density”, Geometry group report, Williams College. Prentice-Hall, Englewood Clids N. J.
[6]. Q. Maurmann and F. Morgan (2009), “Isoperimetric comparison theorems for manifolds with
Density”, Calc. Var. PDE, Number 36, No. 1, P. 1-5.
[7]. F. Morgan (2005), “Manifolds with density”, Notices Amer. Math. Soc., Number 52, P. 853-858.
[8]. F. Morgan (2006), “Myers Theorem with density”, Kodai Math. J., Number 29, P. 454-460.
[9]. F. Morgan (2009), “Manifolds with density and Perelman's proof of the Poincare Conjecture”,
Amer. Math. Monthly, Number 116, P. 134-142.
[10]. C. Rosales, A. Cãnete, V. Bayle and F. Morgan (2008), “On the isoperimetric problem in Euclidean space with density”, Calc. Var. PDE, Number 31, no. 1, P. 27-46.
[2]. R. Corwin, N. Hoffman, S. Hurder, V. Sesum, and Y. Xu (2006), “Differential geometry of manifolds with density”, Rose-Hulman Und. Math. J., 7 (1).
[3]. A. Cannas Da Silva (2001), Lectures on Symplectic Geometry, Spriger – Verlag – New York – Berlin –- Heidelberg – Tokyo.
[4]. M. Gromov (2003), “Isoperimetry of waists and concentration of maps”, Geom. Funct. Anal., No. 13, P. 178-215.
[5]. C. Ivan, H. Stephanie, A. Vojislav, And Y. Xu (2004), “Double bubbles in Gauss space and highdimensional spheres, and differential geometry of manifolds with density”, Geometry group report, Williams College. Prentice-Hall, Englewood Clids N. J.
[6]. Q. Maurmann and F. Morgan (2009), “Isoperimetric comparison theorems for manifolds with
Density”, Calc. Var. PDE, Number 36, No. 1, P. 1-5.
[7]. F. Morgan (2005), “Manifolds with density”, Notices Amer. Math. Soc., Number 52, P. 853-858.
[8]. F. Morgan (2006), “Myers Theorem with density”, Kodai Math. J., Number 29, P. 454-460.
[9]. F. Morgan (2009), “Manifolds with density and Perelman's proof of the Poincare Conjecture”,
Amer. Math. Monthly, Number 116, P. 134-142.
[10]. C. Rosales, A. Cãnete, V. Bayle and F. Morgan (2008), “On the isoperimetric problem in Euclidean space with density”, Calc. Var. PDE, Number 31, no. 1, P. 27-46.
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