Skip to main content
MathOverflow

Return to Question

Typo
Source Link
LSpice
LSpice
  • 12.9k
  • 4
  • 45
  • 69

A manifold whose tangent space ofis a sum of line bundles and higher rank vector bundles

I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition $$ T(M) = A \oplus L_1 \oplus \cdots \oplus L_k, ~~~ k > 0, $$$$ T(M) = A \oplus L_1 \oplus \dotsb \oplus L_k, ~~~ k > 0, $$ where each summand $L_i$ is a line bundle over $M$, and $A$ is a vector bundle that cannot be written as a direct sum of line bundles. If $M$ admitted a nice (compact) group action then that would be even better.

A manifold whose tangent space of a sum of line bundles and higher rank vector bundles

I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition $$ T(M) = A \oplus L_1 \oplus \cdots \oplus L_k, ~~~ k > 0, $$ where each summand $L_i$ is a line bundle over $M$, and $A$ is a vector bundle that cannot be written as a direct sum of line bundles. If $M$ admitted a nice (compact) group action then that would be even better.

A manifold whose tangent space is a sum of line bundles and higher rank vector bundles

I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition $$ T(M) = A \oplus L_1 \oplus \dotsb \oplus L_k, ~~~ k > 0, $$ where each summand $L_i$ is a line bundle over $M$, and $A$ is a vector bundle that cannot be written as a direct sum of line bundles. If $M$ admitted a nice (compact) group action then that would be even better.

Became Hot Network Question
added 12 characters in body
Source Link
Bobby-John Wilson
Bobby-John Wilson
  • 75
  • 7

I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition $$ T(M) = A \oplus L_1 \oplus \cdots \oplus K_k $$$$ T(M) = A \oplus L_1 \oplus \cdots \oplus L_k, ~~~ k > 0, $$ where each summand $L_i$ is a line bundle over $M$, and $A$ is a vector bundle that cannot be written as a direct sum of line bundles. If $M$ admitted a nice (compact) group action then that would be even better.

I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition $$ T(M) = A \oplus L_1 \oplus \cdots \oplus K_k $$ where each summand $L_i$ is a line bundle over $M$, and $A$ is a vector bundle that cannot be written as a direct sum of line bundles. If $M$ admitted a nice (compact) group action then that would be even better.

I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition $$ T(M) = A \oplus L_1 \oplus \cdots \oplus L_k, ~~~ k > 0, $$ where each summand $L_i$ is a line bundle over $M$, and $A$ is a vector bundle that cannot be written as a direct sum of line bundles. If $M$ admitted a nice (compact) group action then that would be even better.

Source Link
Bobby-John Wilson
Bobby-John Wilson
  • 75
  • 7

A manifold whose tangent space of a sum of line bundles and higher rank vector bundles

I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition $$ T(M) = A \oplus L_1 \oplus \cdots \oplus K_k $$ where each summand $L_i$ is a line bundle over $M$, and $A$ is a vector bundle that cannot be written as a direct sum of line bundles. If $M$ admitted a nice (compact) group action then that would be even better.