Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the ighted sum of squared distances to the vertices is the Karcher mean relative to the ights. Using barycentric coordinates as the ights,  obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of this barycentric coordinate map. In this work  articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not,  say the Riemannian simplex is degenerate. Quality measures for the “thickness” or “fatness” of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. Hover, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.
['Ramsay Dyer', 'Gert Vegter', 'Mathijs Wintraecken']
I am writing a research paper on a idea Riemannian simplices and triangulations
[{"from": "human", "value": "I am writing a research paper on a idea Riemannian simplices and triangulations"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the ighted sum of squared distances to the vertices is the Karcher mean relative to the ights. Using barycentric coordinates as the ights,  obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of this barycentric coordinate map. In this work  articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not,  say the Riemannian simplex is degenerate. Quality measures for the \u201cthickness\u201d or \u201cfatness\u201d of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. Hover, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold."}]