How Long to Read Volume Comparison, Ricci Curvature, and Focal Radius

How Long Does it Take to Read Volume Comparison, Ricci Curvature, and Focal Radius?

It takes the average reader 1 hour and 10 minutes to read Volume Comparison, Ricci Curvature, and Focal Radius by Robert James Willett

Assuming a reading speed of 250 words per minute. Learn more

Description

In this paper, we seek to provide counter examples to two volume comparison lemmas found in [3] if we generalize their assumptions to a lower Ricci curvature bound. Second we seek to further understand Riemannian manifolds which contain embedded submanifolds of certain focal radius. First, in papers [2], [3] Grove and Petersen discussed the relationship between bounds on sectional curvature, radius, and volume and its effects on the topology of a closed Riemannian manifolds. They prove that for manifolds with a lower sectional curvature bound, an upper radius bound and almost maximal volume, one can give topological equivalence to either Sn or R Pn. It was later proved to be diffeomorphic [9]. Also, Grove and Petersen showed that the limit space of a convergent sequence of manifolds with maximal volume has certain geometric properties. In the proofs of these theorems they use two powerful volume comparison lemmas. We discuss why the methods used in [3] cannot be extended in general to manifolds with a lower Ricci curvature bound by looking at some interesting counter examples. Second, in a paper by Guiljaro and Wilhelm [4] we see a relationship between closed embedded submanifolds of maximal focal radius and topological type, and seek to understand the necessity of the submanifold being closed by providing counterexamples in which we consider Manifolds with embedded open submanifolds of certain focal radius which do not satisfy the conclusions of the results in [4].

How long is Volume Comparison, Ricci Curvature, and Focal Radius?

Volume Comparison, Ricci Curvature, and Focal Radius by Robert James Willett is 70 pages long, and a total of 17,500 words.

This makes it 24% the length of the average book. It also has 21% more words than the average book.

How Long Does it Take to Read Volume Comparison, Ricci Curvature, and Focal Radius Aloud?

The average oral reading speed is 183 words per minute. This means it takes 1 hour and 35 minutes to read Volume Comparison, Ricci Curvature, and Focal Radius aloud.

What Reading Level is Volume Comparison, Ricci Curvature, and Focal Radius?

Volume Comparison, Ricci Curvature, and Focal Radius is suitable for students ages 8 and up.

Note that there may be other factors that effect this rating besides length that are not factored in on this page. This may include things like complex language or sensitive topics not suitable for students of certain ages.

When deciding what to show young students always use your best judgement and consult a professional.

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