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# Embedding and ingraining science, and building and broadening the capacity for research and development across society ;

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# Gauss – Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ ( M ) where χ ( M ) denotes the Euler characteristic of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem.

# Nash embedding theorems also called fundamental theorems of Riemannian geometry.

# The Cartan – Hadamard theorem states that a complete simply connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space R ^ n with n = dim M via the exponential map at any point.

# The geodesic flow of any compact Riemannian manifold with negative sectional curvature is ergodic.

# If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT ( k ) space.

# The isometry group of a compact Riemannian manifold with negative Ricci curvature is discrete.

# Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.

# If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n ( n-1 ).

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# Myers ' theorem states that if the Ricci curvature is bounded from below on a complete Riemannian manifold by, then the manifold has diameter, with equality only if the manifold is isometric to a sphere of a constant curvature k. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental group.

# The Bishop – Gromov inequality states that if a complete m-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a ball is smaller or equal to the volume of a ball of the same radius in Euclidean m-space.

# The Cheeger-Gromoll splitting theorem states that if a complete Riemannian manifold with contains a line, meaning a geodesic γ such that for all, then it is isometric to a product space.

# Pontryagin numbers of closed Riemannian manifold ( as well as Pontryagin classes ) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.

# Riemannian geometry

# Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 ( real ) dimensions.

* Glossary of Riemannian and metric geometry # R for its meaning in Riemannian geometry.

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