Much of his research concerns the Riemannian geometry of 4-manifolds, or related topics in complex and differential geometry.
32.
These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory.
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Modern geometry has many ties to physics as is exemplified by the links between pseudo-Riemannian geometry and general relativity.
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However, ordinary Riemannian geometry is unable to describe the properties of the electromagnetic field as a purely geometric phenomenon.
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Barrett O'Neill's Semi-riemannian geometry ).
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In Riemannian geometry, the existence of a metric chooses a unique preferred torsion-free covariant derivative, known as the Levi-Civita connection.
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His mathematican contributions include comparison theorems of Laplacian eigenvalues on Riemannian manifolds and the maximal diameter theorem in Riemannian geometry.
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It's a nice springboard into Riemannian geometry, should you decide to take that at a later date, or general relativity.
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Using Riemannian geometry Holland formulated this method in very general terms that includes as special cases quantum many-particle systems and spin.
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Here is a short list of global results concerning manifolds with positive Ricci curvature; see also classical theorems of Riemannian geometry.
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