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### Is a smooth immersion $c: [a,b] \to M$ injective if its image is a 1-manifold with non-empty boundary?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. In this question, it is confirmed that the example is an error. ...
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### What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under these maps?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. In this question, Prof Jack Lee says that the example is ...
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### Pushfoward of smooth vector field is smooth?

My books are Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An ...
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### Does the Riemannian metric induced by a diffeomorphism $F$ exist for a reason other than the existence of vector field pushforwards?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...