Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Conditions for Nonnegative Curvature on Vector Bundles and Sphere
  Bundles   This paper addresses Cheeger and Gromoll's question of which vector bundles
admit a complete metric of nonnegative curvature, and relates their question to
the issue of which sphere bundles admit a metric of positive curvature. We show
that any vector bundle which admits a metric of nonnegative curvature must
admit a connection, a tensor, and a metric on the base space which together
satisfy a certain differential inequality. On the other hand, a slight
sharpening of this condition is sufficient for the associated sphere bundle to
admit a metric of positive curvature. Our results sharpen and generalize
Walschap and Strake's conditions under which a vector bundle admits a
connection metric of nonnegative curvature.

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The categories of this paper are: q-bio
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin